DOCUMENT RESUME ED 194 317 SE 033 066 Mathematics

DOCUMENT RESUME

ED 194 317 SE 033 066 TITLE Mathematics Evaluation Procedures K-2. For Use by Teachers With Children in Years K-4. INSTITUTION Australian Council for Educational Research, Hawthorn. FEPORT NO ISBN-0-85563-210-0 PUB DATE Sep BO NOTE 279p.: Not available in hard copy due to copyright restrictions. Contains occasional marginal legibility. Prepared in collaboration with the North Sydney Region Infants Mistresses Council. AVAILABLE FROMLawrence Verry, Inc., River Road, CT 06355 (no price quoted).

EDRS PRICE MF01 Plus Postage. PC Not Available from EDRS. rESCEIPTORS Academic Achievement: *Cognitive Measurement; Diagnostic Tests: Elementary Education: *Elementary School Mathematics: Evaluation; *Evaluation Methods: *Learning: *Mathematics Instruction; Resource Materials: Student Characteristics; *Testing ABSTRACT This book was written in response toa need expressed by Infant's Mistresses in the North Sydney Region of the NewSouth Wales Department of Education. They felt therewas a need to evaluate children's progress in mathematics in order to diagnose thecauses of individual failures, and they discovered there was no readily available information on either what or how to test. A committeewas appointed to investigate and rectify the situation. Goals selected were a massive list of learning objectives, and approximately five test items. per objective were constructed. This document consists of instructions and information cn testing, selected notes and comments, the tests, an appendix on trialling, and sample studentresponse sheet forms, the product of several years of work. The testsare devised for administering to individual elementary schoolpupils in grades K -4 with the object of observing theresponses of those pupils. The tests are set out in continuous form withoutage or grade labels, and are organized within three broad categories ofnumber, shape, and measurement. (MP)

*********************************************************************** * Reproductions supplied by EDRS are the best that can be made * * from the original document. * *********************************************************************** Published by The Australian Council for Educational Research Limited Radford House, Frederick Street, Hawthorn, Victoria, 3122

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National Library of Australia Cataloguing-in-Publication data. Mathematics evaluation procedures, K-2. Previously published, North Sydney, N.S.W.: North Sydney Region Infants' Mistresses' Council, 1979. ISBN 0 85563 210 0 1. Number conceptAbility testing. 2. Education, Primary. I. North Sydney Region Infants' Mistresses' Council. II. Australian Council for Educational Research. 372.7'2044

No part of this publication may be reproduced in any form without permission fro the publisher.

II ACKNOWLEDGEMENTS

The North Sydney Region Infants' Mistresses' Council acknowledges the assistance of the following people and thanks them for their efforts.

Mr. Tom Reid, B.A., Director of Education, North Sydney Region, who has given his support and encouragement throughout this project.

Mr. Ted King, M.A., Inspector of Schools, Chairman of North Sydney Region Development Committee, whose help in making In-service facilities available has been invaluable.

Mr. Sid Bourke, B.A., B.Sc., Litt.B., Chief Research Officer, Australian Council for Educational Research, whose guidance and advice have been greatly appreciated.

Dr. Bob Phillips, B.A., Ph.D.,-Head of the Department of Teaching and Curriculum Studies, Alexander Mackie College of Advanced Education, who began the whole project with his talk to our conference in 1974, and maintained a continuing interest in the project.

Mrs. Elaine Callan, Mrs. Marie Muir and Miss Kath Yabsley, Committee members, who have been primarily concerned in selecting the evaluation goals and compiling the test items, also Mrs. Jan Hodge and Miss Jan Watt who were part of the committee in the early stages when the goals were being listed.

The people who were involved in putting together the Trialling Document, all those who typed, duplicated and collated, and especially the staff at Alexander Mackie C.A.E. who stapled and bound all the copies for use.

All the mathematicians, educationists and college lecturers who read, considered and checked the draft document and gave the committee the benefit of their specialised knowledge, particularly Mr. Alex Aitkin, M.Sc., formerly of Sydney Teachers College, whose thorough perusal and informative comments were much appreciated, Mrs. Nita Hughes, M.A., Dip.Ed., Senior Lecturer, Mathematics, Primary Programme Director, Sydney Teachers College, who made many pertinent comments on the text, and Miss Joan Sutton, B.A., M.Ed., Inspector of Schools, who gave help in assessing conservation activities.

Those who sent to the Committee their written comments on the draft document:

Mr. Eric R. Ashcroft, MA., Dip.T., Senior Lecturer, Education Department, Massey University, Palmerston North, N.Z., Mr. W. Barry, M.A., Head of Mathematics, Alexander Mackie College of Advanced Education, Ms. B. Dunbar, B.Ed., Central Metropolitan Region Primary Mathematics Consultant, Professor Wilbur H. Dutton, A.B., M.A., Ed.D., (Emeritus 1948-197-5) Department of Education, University of California, Los Angeles, U.S.A., Mr. W.L. Howard, B.A., M.Ed., Newcastle College of Advanced Education, Mr. Peter Jeffery, B.A., B.Ed., Senior Advisory Officer, Education Advisory Services A.C.E.R., and Mrs. Irene Sawczak. who assisted Mr. Jeffery.

iii The mathematics students of Alexander Mackie College of Advanced Education and of Sydney Teachers' College, who carried out assignments on the test procedures and whose insightful comments were most helpful, also their lecturers, Bob Phillips and Nita Hughes, respectively, who guided them in their work.

Mrs. Joan Graham, w helped the Committee during the early stages in the formulation of the goals.

Dr. Robert Perry, B.Sc., Dip.Ed., Ph.D., Mitchell College of Advanced Education, who co-ordinated trialling in the one-teacher schools of the Bathurst area.

Mr. Kevin Earl, Publications Officer, Alexander Mackie College of Advanced Education, who gave helpful advice on publication procedures.

The Mistresses, Principals and class tecchers of the schools where trialling was carried out, who worked on the tests, recorded their comments, met with the committee, offered their suggestions and gave their opinions, and so shaped the final form of the book.

Lastly all the children who were the "guinea-pigs" for the trial testing, whose responses were diagnosed, and whose progress was analysed, so that many other children may in the future be helped to mathematical competence through use of these tests.

iv CONTENTS Page Acknowledgements iii

Index of Objectives

Number vii Shape Measurement

Introduction 1

Instructions and Information

Testing 3 Approach 3

Purposes 3 Format 4 Materials 5 Administration 6 Assessment 7 Recording Responses 8 Suggested Form for Response Sheet 9

Filling in a Response Sheet 11 Interpretation of Marking 12 Follow-up 12

Notes and Comments

Advantages of Testing 13 The Time Taken in Testing 14

Common Causes of Failure 15 Testing Beyond Grade Level 16 Suggested Grade Levels for Diagnostic Testing 16 Memorization, Problem Work, Graphs 19 Use of "Crutch" Figures 19 "Number" Bias 20 Conservation 20 Geheral Knowledge 21 Prompting 21 Wrong Responses 22 Notes and Comments (continued) Page How Many Times? Size of the Book 23 Why "Sets" and "Subsets"? 23

The Tests Number 26 Memorization 113 Problems 128 Graphs 148 Shape 156 Measurement Length 169 Mass 185 Volume (Capacity) 198 Area 212 Time 221 Money 231 Temperature 239

Appendix Outline of Trialling Feedback 242 Documents 257

Response Sheet Forms 263

vi INDEX NUMBER

No. OBJECTIVES. Test items begin The Child ... on page

NUMBER.

1. Identifies attributes 26

2. Classifies an object by means of its attributes 26

3. Sorts objects 28

4. Makes given sets

5. Pairs the elements of one set with the elements of another set, and finds whether two sets match or do not match by using the technique of pairing 30

6. Understands that two equivalent sets remain equivalent however the members of each set are re-arranged 32

7. Counts the members of a set 33

8. Finds the cardinal number of a sei 33

9. Makes a set containing a given number of elements 34

10. Matches numeral with cardinal number 35

11. Writes numeral for cardinal number of a set 37

12. Uses ordinal numbers 37

13. Orders sets (counting order) 40

14. Understands the relationship of one set to the next in the counting order 41

15.Understands that a set has the same cardinal number when the members are re-arranged 42

16. Understands and uses the terms "more" and "less" through comparison of sets 43

17. Understands that all sets which arc equivalent have the same cardinal number 47

18. Understands the meaning of, and writes, the equals sign 50

19.Separates a set into subsets and understands the part-whole relationship . 53

20.Knows that zero is the cardinal number and nought is the numeral for the set that has no members 58

21.Combines two sets to form a new set and understands that this is an action of addition 60

vii INDEX NUMBER

No. OBJECTIVES. Test items begin The Child.,. on page

NUMBER (continued).

11 Understands that whatever the order in which sets are combined, the total number of members is the same 63

23. Separates a set into two subsets. removes one subset and understands that this is an action of sohtraction 64

24. Compares two sets by pairing to find the difference between them, and understands that this is also an action of subtraction 67

15. Understands that there is an inverse relationship between the operations of addition and subtraction, and of subtraction and addition 69

26. Understandsthatfrom one additionfact,other additionfacts and subtraction facts can be derived 71

'7. Combines several sets each with the same cardinal number to form a new set and understandsthatthisactionofrepeatedadditionis multiplication 74

'8. Partitions a set into two equivalent subsets and understands that this is the action for finding halves 76

29. Understands and uses the terms "odd and "even'' 79

30. Partitions a set into three, four., five equivalent subsets, and understands this as the action for finding thirds, quarters, fifths 84

31. Partitions a set into a given number of equivalent subsets to find out how

many in each subset. and understands that this is the action of sharing . 86

32. Partitions a set into equivalent subsets of a given number to find out how

many subsets are formed, and understands this as an action of division . 90

33. Removes from a set equivalent subsets ()la given mini- er to find out how many times this can he done, and unLrstands that this repeated subtraction can give a division answer 92

34. Records (in number sentences using the signs + x -=-)the actions for operations 95

35. Understandsthevalue of the written symbols representing numbers beyond ten (to99) 110

36. Knows how to write the numerals for given numbers heymid ten (to99) . 113

37* Uses appropriate memorised number facts 113

38* Applies the knowledge gained and the number facts learnt, to composing environment al narratives 128

viii INDEX NUMBER

No. OBJECTIVES. Test items begin The Child ... on page

NUMBER (continued).

39* Applies the knowledge gainial and the number facts learnt, to answering questions contained in given narratives about the environment 142

40*Applies the knowledge gained at id the number facts learnt to representing everyday situations, recording these in a variety of ways, and interpreting the records 148

41* Uses mathematical voc..Julary: factor, product. etc. 155

OPERATIONS.

Addition Objectives: 16. 18, 21, 22, 25, 34 (recording). 37* (memorisation). 38* and 39* (problems).

Subtraction Objectives: 16, 18, 19, 23, 24, 25. 26. 34 (recording), 37* (memorisation). 38* and 39* (problems).

Multiplication Objectives: 27. 34 (recording). 37* (memorisation). 38* and 39* (problems).

Fractions Objectives: 28, 34 (recording). 37* (memorisation). 38* and 39* (problems).

Divisiop: Objectives:21 (sharing),32,33, 34 (recording), 37* (memorisation). 38* and 39* (problem:;).

* Testing inthese Objectivesis to be conducted at the approprivte time in the teaching sequence. and should not he postponed until after testing of Objective 36. INDEX -- SHAPE and MEASUREMENT

No. OBJECTIVES. Test items begin The Child ... on page

SHAPE

i. (a)Identifies attributes and (b) uses vocabulary to describe lines and shapes 156

2. Classifies shapes 157

3. Sorts shapes 161

4. Describes the properties of the square, triangle, circle and oblong !64

5. Applies knowledge of shape in environmental situations 167

6. Uses special vocabulary cy shape 168

MEASUREMENT: LENGTH

1. Compares (a) height and (b) length of two objects 169

2. Seriates by comparing three or more objects (a) of different heights (b) of different lengths 171

3. Measures with informal units 173

4. Esti -Imes length using informal units, with reasonable accuracy 174

5. Measures, choosing the appropriate unit and tool 175

6. Realises the need for a standard unit 176

7. Measures with a standard unit 177

8. Estimates length .,sing standard units 179

9. Knows relationship between centimetre and metre 180

10. Conserves length lql

11. Applies knowledge of length in environmental situations 184

12. Uses special vocabulary of length 184

MASS

1. Compares the mass of objects through lifting, pushing, pulling 185

2. Understands that size or shape does not dc.ermine mass 186 INDEX SHAPE and MEASUREMENT

No. OBJECTIVES. Test items begin The Child on page

IASS (continued)

3. Understands the use of the beam balance 186

4. Measures mass using informal units 188

5..Sedates by comparing the mass of two, three or four objects by means of the beam balance 189

6. Estimates mass using informal units, with reasonable accuracy 190

7. RealiSes the need for standard units 191

8. Measures with standard units 191

9. Estimates mass using standard units. with reasonable accuracy 192

10. Conserves mass 193

11. Applies knowledge of mass in environmental situations 196

12. Uses the special vocabulary of mass 197

rOLUME (CAPACITY)

1. Understands the terms "full" and "empty" 198

2. Measures capacity using informal units 199

3. Compares the capacity of containers 200

4. Seriates three or mo-e containers according to capacity 203

5. Estimates capacity using informal units with reasona!)le accuracy 204

6. Understands conservation in relation to capacity 205

7. Realises the need for a standard unit 207

8. Measures with a standard unit (litre) 208

9.Estimates capacity using a standard unit (litre), with reasonable accuracy 209

10. Applies knowledge of capacity in environmental situations 20Q

11. Uses special vocabulary of capacity 210 10 xi INDEX SHAPE and MEASUREMENT

No. OBJECTIVES. Test items begin The Child ... on page

AREA

1. Understands the term "covering a surface" 212

2. Covers a surface using informal units, and discovers the number of units required (i.e. finds the area) 214

3. Chooses a unit appropriate to the area of the surface to be measured 215

4. Finds shapes that tesselate to make a surface 216

5. Conserves area in terms of informal units 217

TIME

1. Understands the time divisions of a day and relates these to daily activities 221

2. Tells the time from "traditional" and digital clOcks, and relates times to daily events 222

3. (a) Knows the days of the week, in sequence, (b) knows the months of the year in sequence, (c) uses a calendar and (d) writes dates 224

4. Understands elapsed time, and uses related vocabulary 227

5. Applies the knowledge of measurement of time, using standard units, to environmental situations 229

6. Uses special vocabulary of time 230

MONEY

1. Recognises coins 231

2. Knows the value of the coins 232

3. Counts coins to find the amount of money 233

4. Uses appropriate coins for purchasing 233

5. (a) Understands the concept of change and (b) carries out shopping transactions 234

6. Recognises the dollar note and the two-dollar note 236

7. Reads and uses symbols for recording amounts 237

8. Understands the need for money 238 is INDEX SHAPE and MEASUREMENT

No. OBJECTIVES. Test items begin The Child ... on page

TEMPERATURE

1. Recognises differences in temperature 239

2. Uses vocabulary related to temperature 239

3. Applies knowledge in environmental situations 239

* INTRODUCTION

This book was written in response to a need which no doubt could be expressed in terms expressed by Infants' Mistresses in the North of an objective. The Committee looked at Sydney Region of the N.S.W. Department of numerous such sequences of learning-steps; Education. They felt that there was a need it was felt that often the objectives in these to evaluate children's progress in sequences merely outlined teaching Mathematics in order to diagnose the causes strategies, or referred to intermediate stages, ofindividualfailures.Therewas,they or described discovery techniques; generally discovered, no readily available information they marked very small advances in skills. as to what to test and how to test for this Most did not constitute necessary founda- purpose. tionsforthenextlearning stage. The Committee wasfacedwith thetask of They therefore appointed a Committee to selecting, from these detailed lists, a series investigatethesituation andif possible of major objectives which would warrant rectify it. evaluation to identify levels of achievement. This Committee was set up in October, The committee spentalmost two years 1974.Its members were Mrs. E. Callan, selectingthemathematicalgoals.These Mrs. M. Muir, Miss K. Yabsley, Miss J. Watts covered Number, Shape and the various and Mrs. J. Hodge. strandsof Measurement (Length,Area, Money, Mass, Temperature and Volume). Its terms of reference at that time were: The goals were then expressed as Objectives "To detail inathematical goals, so that for a child to reach and master. In each thoseskilledinthetechniques of strand, these Objectives formed, as far as measurement can formulate evaluation ispossible, a logical sequence. The ones procedures.Itisanticipatedthat selected constitutedthe important steps, teachers will then be able to identify the "goals" of achievement, ina child's the levels of understanding reached by progress.Theintermediatestepswere individual children." omittedintheevaluationprogramme, though of course they would not be omitted This was consequent on an address to the from the teaching programme. North Sydney Region Infants' Mistresses' Council (N.S.R.I.M.C.), by Dr. Bob Phillips, An example of how the selection was made who spoke of the necessityfor, and the is afforded by the work in Area. A suggested difficulties of, effective evaluation in the learning sequence was: (a) understands the teachingofmathematicstopics.Those terms "covering" and "surface", (b) com- present were mainlinterested in the under- pares the surface of objects by placing one standings, skills and knowledge set out in over the other, (c) senates objects according the N.S.W. mathematics syllabus for infants' to their surface area, (d) covers a surface with grades. A workshop-type follow-up plainly informal units and counts them, (e) under- demonstrated to all present the impossibility stands and uses the term "area", (f) finds of carrying out any effective assessment area in terms of informal units, (g) chooses a withoutfirstformulatingspecificgoals. unitappropriatetothesurfaceto be Dr. Phillips indicated that when practising measured, (h) estimates area in terms of teachers had compiled these goals, those informalunits,(i) Itwas "skilled in measurement" could set about decidedthatcomparing,seriatingand composing the necessary test items. estimating were but intermediate learning stepsin Area at this level, and did not In anylearningsequencethereisa constitutegoals,hencetheywerenot progression of many smallsteps, each of included as major Objectives to be evaluated. When the Committee had completed the Itwas indeed,asMr. Bourkeforecast, compiling of the goals inallthe Strands, "complex and demanding". A lotcif care they looked about for someone "skilled in and thought have heen put into it. This is measurement techniques" to carry out thc. the result. secondpartof thetask:"toformulate evaluation procedures". Meetings were held The work was complex because there are so withvariousDepartmentalOfficersand many strands, each with numerous other experts including Mr. Sid Bourke of important goals.It was demanding because theA.C.E.R.Theoutcomesofthese diagnostic procedures had to be described meetings werethatno one"skilledin for each separate goal atleast one per measurement" was available, and that those goal, usually several for each, to cater for consulted consideredthatthe Committee different aspects and different levels. The members themselves should undertake the size of the resultant volume may be dismay- task, as they were familiar with the goals ing to readers unless they realise that itis specified, and with classroom procedures large because itis comprehensive, and itis in Infants' schools. comprehensive sothatinit, somewhere, they may findtheparticularassistance Mr. Bourke stated (October, 1976): "The they need. lists of mathematics goals devised by the Committee are very comprehensive and can Thetestshavebeenusedinselected be used as a basis for determining the skills schoolsforatwelve-monthtrialperiod and knowledge possessed by children. The (See Appendix). task of writing approximately five items for each goal is a very large one indeed, but one Itis hoped that this document will be of which could have considerable benefits for value to teachers in their work with children, users In my opinion, the members and that in some small measure it may assist of the Committee are best placed to write them to improve their planning of learning the required items. Any other group would experiences, and their knowledgeof have to spend considerable time and energy children's learning difficulties.In shortit goingovermuchofthe work already is hoped thatit may help teacheis to help completed by the Committee in stating goals children. before they would be ready to commence developing assessment items

The Committee therefore set about the task. December, 1978.

16 INSTRUCTIONSAND INFORMATION TESTING competency. Some of them relate to understandings, some to skills and some to the Testinginconnectionwith Mathematics assimilation of knowledge. Each of them is Evaluation may be carried out for any one considered anecessaryfoundationfor of a variety of purposes. further progress. There are numerous intermediate goals occurring between those given There are tests which are organized to here, but they do not warrant any intensive rank a givensetof pupils,in order, evaluationsincetheymerelyserveto from the most competent to the least indicate necessary learning steps. along the competent. way.

There are tests which are designed to The Objectives are not listedin strictly ascertainif pupils have reached a pre- hierarchical order. The order suggested is scribed standard of achievement. reasonably sequential. An absoluteorder cannotbeprescribedbecause(a)"all There are tests which are standardised to children are different", (b) children learn provide a norm for comparison of the things indifferentsequences,and(c) achievements of particular classes or of children learn things concurrently. individual pupils. For the purpose of programming lessons, There aretests which are devised for or testing at varying age-levels, or catering administering to individual pupils with for intermediate stages, the Objectives may the object of observing the responses of be broken into sub-objectives. those pupils. Forexample,theObjective"recognises The tests in this book are of the last-named attributes" can if desired be narrowed to type and atedesigned primarily so that "recognises colours", then to "recognises teachers may help children. red". The Objective "knows how to write numerals for numbers beyond ten" can be presented in a series " from ten to APPROACH twenty," " from twenty to thirty," from thirty to forty" and so on. As explained above, the tests set out in this book are for the purpose of helping teachers TheObjective"coversasurfacewith toevaluatepupils' work, sothat where informal units,discoversthenumber a child is not succeeding, help may be given required and finds the area" may be broken through the diagnosis afforded by the tests. into itsthree parts to become three sub objectivescateringforthreedifferent The approach to diagnostic evaluation in age-levels "covers a surface with informal this book is through a series of Objectives units", "counts the number of informal on which the sets of tests are based. units covering a surface" and "uses informal units to find the area of a surface". Each Objective is stated in terms of a child's achievement,ratherthanof ateacher's lesson aim. PURPOSES

The particular Objectives used have been The tests may be used by a teacher selected because they are important goals *as a means of assessing the understand- in achild'sprogresstomathematical ing of an individual class member, or

11-1 3 *for diagnosing the area of difficulty of The total number otests and test items is a non-achiever, or considerable. Series of tests are provided for *for discovering the level of competence Objectives in the various strands: of a new-comer, or Number (41 Objectives), Shape (6), Length *asameans of self-appraisalof the (12), Mass (12), Time (11), Money (8), classroom teaching. Area (5), and Volume (11).

In addition, for the class teacher, the list of For each child tested only a portion of these Objectives may be helpful in programme- will be used. No child will need to be given making, and perusal of the test procedures all the tests in all the strands. may suggest the kinds of experiences which can help to ensure understanding. A reading It cannot be too highly stressed that in the of all the Objectives may alert the teacher to name of evalu Rion (be it of child, or class, areas of instruction which have been omitted or school) itis not intended that any child from the teaching sequence. be subjected to the whole barrav of tests outlined here. Only those tests should be The tests however are primarily for purposes given to a ctild which are necessary to of diagnosis. That is, they are designed to discover his level of attainment, his degree help teachers to help children. Children fail of understanding, or his area of difficulty for different reasons, and it is difficult to and, later p rhaps, the amount of improve- give assistance until the reason for the failure ment that has taken place. is discovered. The tests are not intended as standardised *A child may be failing because of a procedures for comparing children. Nowhere developmental difficulty. isit suggested that provision be made for *He may be a newcomer whose previous recording achild'stotalmarks; such a level is unknown. score would be quite meaningless in relation *He may fail because of a lack of teach- to a child's ability or for comparison with ing in a certain aspect of the work. any other child's ability. *He may have failed because he did not assimilate the teaching that was given. *He may not be familiar with certain vocabulary. FORMAT *He may lack practical experience of the material or the activity in question. The tests are set out under Strand headings, firstthe Number strand, (which includes Problem Solving and Graphs), then the Thetests are designed to help teachers Shapestrand,andlastlyMeasurement identify the area and cause of difficulty, so (Length, Mass, Volume, Area, Time, Money that a programme of help may be arranged and Temperature). for the particular child. Sets of tests are described for each Objective T1-.easpect of mathematics which causes in each strand. most concern to the class teacheris of course that of Number. However the other They are arranged in a format designed to be aspects also are of importance, and early read easily so that the teacher may follow experiences can influence later development; the procedures with minimum difficulty. early misunderstandings or omissions nay have a bearing on later attitudes or diffi- Each test is set out in three columns. The culties. Hence there are test items prepared first states the Objectives being evaluated. for these other strands also, so that evalua- The second lists the materials which will be tion may, if and when necessary, be carried needed for the pupil's use; some of this outin any oralltheother aspects of material must first be made, most of the mathematics. It is hoped that this will make items are readily available and merely need the book more valuable to teachers, even to be assembled and organised. The third thought it makes it much lengthier. column sets out the test procedure steps in _18 detail, using an instruction form: "Place ..." example, in Objective 2 of the Number "Say: ..." "Ask: ...". strand,thetestprovidesforquestions covering shape, colour, texture, size, usage, Guidance is given as to the kindf response and so on, in the same assessment. The whichwill indicatethatthechildis teacher may however, if a more detailed competent in the situation. Sometimes there assessment is desired, provide a separate test is information about a response which will of five or more questions for shape, then a point to an intermediate stage of develop- separate test of five or more questions for ment. Beforeassessingthequalityof a colour, and so on. child's answer the teacher must refer to the Acceptable Response statement given for Itwill be noted that in some cases the each test procedure. procedure outlined is more lengthy than in others. It seems to be a long time before one One correct response is rarely sufficient to arrives at the test question. The stage, as it assume that a child has no difficulties. It is were, hasto be setfor the test. These necessary to repeat the test several times in preliminarystepsareimportant' forthe slightly varied form to ascertain if the child child's understanding of the test, but his is consistentlysuccessful.Accordingly, responses here are not to be recorded as part provisionis made inthe instructions for ofthetest.Incases such asthis,the suitablerepetitions oftf,etestsineach beginningof thetestproper isusually separate procedure. marked with a double asterisk, thus **. For example,Objective17inthe Number It is suggested that, at any one level, at least strand concerns the set with no members. five andprobablyten,repetitionsare It is not reasonable to present a child with a necessary for adequate assessment. In some bare table-top and expect him to recognise of the test outlines, five examples of each this as an example of the empty set. It is procedure are set out. In others, only one necessary to show him a set whose members is set out, and the teacher must frame at he recognises, andallof which he sees least four others of a similar type (generally removed at this stage the bare surface will by varying the numbers of objects used, or have meaning for him. the types of objects selected), so that there are sufficientresponses on which to base The pupilisreferred to throughout as a a judgement of thechild's consistency. male child, and the teacher as female. This There may be times, especially with lengthy ispurely for com -nience in writing and procedures, when one good performance reading, to avoid cumbersome forms such as may be considered sufficient. "he/she" and "his/her".

Each part of a test is called, in this series, a "test item". A test may consist of five or MATERIALS ten test items. A child is not considered to have succeeded inatestunless he has In each of the strands the materials used for answered correctly an agreed number of testing are largely those found in classrooms; test items. some are easily made by teachers, and a few are available commercially. On occasions, where five items (instructions or questions) are set out, the teacher may For the Number strap;', the majority of the discern variations in the range of difficulty. tests are set out using objects found in the Eecause of the teaching procedure involved child's surroundings. These may be environ- andthetime-spanallottedtoit,these mental objects such as pencils, spoons, toys; questions areconsidered suitablefor the or they may be classroom objects such as level of understanding under review. In cases round counters or plastic beads. The use of where the child fails in one aspect of the suchmaterialsisvital because a child's test (or where the teaching range has been notion of number comes from his knowledge narrowed) the teacher may frame five or of his environment. and the purpose of more questions based on each aspect covered learning about number isto give under- in the suggested procedures outlined. For standing which can be applied to solving

Jc 5 problems that will occur in the course of Thetestsaredesignedasface-to-face his interaction with his environment. activities in a one-to-one situation as itis felt that only in this way can an adequate A child however must not only discover diagnosis be made. In addition to assessing number facts and understand their signifi- correct responses in such a situation, the cance,he must alsoorganise, remember teacher may be assisted in her evaluation by and apply his knowledge of these number noting a child's reactions or pace of working. facts. In many classrooms, parallel with the Even written work in the testing of opera- recognition of the environmental basis of tionsis best carried out and observed in mathematical structures, children are taught a one-to-one situation. When a written test to codify and tabulate number facts using has been completed, interview-type follow- structured material (because of the economy up is often advisable, and can reveal reasons of timethis ensures, and the consistent for a child's responses often a child may model the material provides). For this reason giveacorrectresponsefor an incorrect ithas been thought desirable to include reason, or perhaps an incorrect response tests to probe the child's understanding of after correct reasoning. work given using this material, and of its relationship tosimilarexampleswith The materials required for each test should environmental objects. Such tests of course be prepared and assembled before the test are only intendedfor children who are begins so that there are no time-wasting learningtousecontinuousstructured distractions. The tester should have read material, such as Cuisenaire rods, for finding the test items and should know the accept- and using number facts. For other types of able responses before beginning, so that the structured material such as centicubes or test can proceed smoothly. Provision for unifix which can be made into rods the recordingthechild'sanswers should be procedures as set out may be adapted, or made, and any papers arranged sothat similar tests devised, in order to test the marking can be easily carried out. A relaxed children's knowledge of their material. The stimulating atmosphere is desirable. intended application of these tests may be stated colloquially: For thetests, theteacher's instructions, directions and questions as given are suit- *Only "rod children" should do the rod able, but are to be taken as suggestions only. tests. According to circumstances, the teacher may *Allchildrenshould do the non-rod find other words more appropriate or more tests. likely to produce responses from the child. A childis often more responsive with his This may be stated in another way: own teacher's familiar mode of expression; itis necessary however to ensure that the *Thenon-rodtestsaredesignedfor changed language does not alter the nature allchildren. of the questions, nor provide the child with *The rod tests are only to be given to undue clues. "rod children". *The rodtestsareadditional tothe Since the tests are not intended as stan- non-rod tests. dardised procedures for comparing children, and because they are designed to evaluate the progress of an individual child,itis ADMINISTRATION permissible to change the suggested language or extend the suggested questions to probe The tests are designed to be given to one whether understanding is present inthat child at a time, ideally in a quiet spot away child. Sometimes an intermediate stage of from the rest of the children. When this is development may bediscernedthrough not possible then the tests should be given such extensions. in group-work sessions, the teacher carrying out the test procedures with one child only, Justasthe wording may be changed if the other children working independently considered desirable, so the materials used at some other task. may be altered according to availability.

206 The suggestions about objects for compiling Objective37;problem-worktestingis sets (blocks, beads, toys and so on) are to be intendedtobeslotted into appropriate taken as suggestions only and other objects places in the teaching sequence following may be substituted if more convenient; the studyoftherelevantoperationsand framing of teacher's questions will then be numbers, at appropriate levels of difficulty. changedtomatch. Frequently a'so,the number of objects suggested is an example Further, a teacher may desire to assess the only. For instance, for "place two equivalent development of the concept of Conservation sets of eight members each", the eight may in a particular child, and may select a series be varied as desired. In such circumstances of tests taken from parts of various strands. the text is often marked ft. Such variations from the printed sequence There may be initial difficulty in ascertain- are not only permissible, but desirable and ing the starting point for testing z child. A necessary. number of alternativesarepossible. The teacher may be guided by the present level of the other children in her class; she may ASSESSMENT startata point she thin':s will confirm a suspected weakness; or she may decide to Assessment of a child's performance is made begin with objectives known to be attained through observation of his reactions during by the child and "wo;k up" to his level of testing, and analysis of his responses to the difficulty, perhaps even skipping a test or testquestions. A recordofthechild's twoifthetentativestartingpointhas answers must be kept so that his level can proved to be too low. Sometimes it may be be assessed, by means of a diagnosis made desirable to begin at a stage considered to be from his successes aid failures. too easy in order to establish confidence. With a very young or a very backward child A separate assessment must be made for she may simply decideto beginatthe eachObjective.Successesindifferent beginning. However, once a firm starting- Objectives cannot be compared or added point has been established within a strand, together. In some cases an Objective is tested itis intended that tests be given consecutive- withonly one setof Procedures asin ly as far as possible, and not in mere random Number, Objective 20; in others there are order. The sequence of tests will be con- many tests, as in Number, Objective 37, tinued until repeated failures occur, which where there are about 100 separate sections. will indicate the child's level of achievement A combined score made up of the results or his stage of development or the cause of from testing more than one Objective could his difficulty. not have any meaning where such widely differing numbers of questions are asked. Although the tests have been listed far as Nor can the value of one Objective be com- possible in a logical sequence, this sequence pared with that of another by this means. is not necessarily the learning sequence. For some Objectives several levels of understand- Each Objective is tested with at least one ing or achievement may be set out. When Procedure, usually sevcal. Each Procedure thisoccursthe teacher must select the sets out a number of variations, called test appropriate level at which to test a particular items, to which the child must respond. A child. teacher should not assume that a child is necessarilycompetentafter one correct This applies particularly when testing know- response and so discontinue the testing for ledge of appropriate vocabulary in the various that Objective. strands, and when testing memorization of number facts as outlined in Objective 37. The question of the number of correct responses required for the assumption of Similarly all the tests for competence in competency must be considered. Probably Problem work are outlined iri Objectives 38 eight out of ten could be taken as conclusive and 39. Itis not intended that testing of for reasons of time-saving, four out of five this work be delayed until after treatment of may frequently be considered adequate.

7 9 Sometimes five out of five (i.e. none wrong) Because of the large number of Objectives or nine out of ten (i.e. one wrong) may be a and the differing numbers of Procedures better requirement to indicate competence. given for each, and also the fact that all the Procedures will not be given to each child Each successful response must be noted, or tested, it will not be practicable to draw up each unsuccessful one, in order to asses forms on which the Objectives and items are competence. Guidance is given under the already set down in print. A form with heading "Acceptable Response" as to the "blank" columns where the Objectives can type of answer which may be marked as be "written in" bytheteacherwill be successful for each item. This should not be adequate; the pages can be numbered and assessed untilthe "Acceptable Response" stapled together if a child needs more than section has been read by the tester. It should one sheet to record all the tests necessary be noted that for some items a number of for diagnosis of his problem. The type of questions, not merely one, must be answered Response Sheet decided upon should be correctly before the response can be con prepared on a stencil-master and duplicated sidered acceptable. For example, in Number, as required so that there will be uniformity Objective 6, the child must maintain equiv- of recording within a school. alenceofsetsthroughanumber of re-arrangements. There will need to be columns where the following details are noted: the Objectives It is desirable that the teacher's assessment being evaluated, either written out in full, be not an"on-the-spot" judgment, but or abbreviated in some way; the particular recorded in some permanent form, for con- Procedure being used; the level at which it is tinued reference, so that a child's progress being applied if there are several levels; the can be monitored over a period of time. exact number of acceptable or unacceptable responses a child makes; lastly a comment Each teacher may devise her own record of a as to whether the overall performance for child's answers and may note thereon the thesetestitemsis considered successful. Objectives tested, the number of successful Perhaps also, notes may be required as to responses made, followed by her diagnosis the quality of the child's answers, or his of the child's performance, together with her attitudes durhg the testing. On the sheet, programme for assistance to the child. It is, too, other particulars will probably need to however, desirable that such a record of be recorded,for example, the date, the information about a child be available to child's age and class, perhaps after the test his later teachers, and in a form which can the assessment of his stage of development, be interpreted easily by them. Hence, within or the diagnosis of his difficulty. There may a school, a record paper with a uniform be a space for the tester's name; alternatively marking codeisdesirable,and with a the school, particularly if large, may require uniform format. thata teacher signa record-sheet before handing it on to the next class teacher.

RECORDING RESPONSES The columns on the sheet, may be arranged verticallyorhorizontallyaccordingto Itis necessary as previously stated that a preference. record be made of a child's responses to the test items for assessment of hislevel of A standardised marking code within a school understanding.It isdesirablethatthis is also useful. It is desirable that the marking record be retainedfor study, so that a code agreed upon in the school to specified diagnosis can be made and a programme and noted on each sheet. Each tester in the developed to meet the child's continuing school can then use the same code, so that needs. Each teacher can if desired design allwho subsequentlystudythechild's and maintain a record book for use in her performance as recorded on the sheet will own classroom. Itwill however be more be able to interpret the marking. helpfulfor those administering and inter- pretingtestsifastandardisedform of A speedy and unobtrusive method of noting recording is used within a school. acceptable and unacceptable responses, and

22 8 the number of each, will be the. main con- This method will cause no ambiguity, sideration. When testing is incomplete for since the only duplication of initials is a particular Objective it may be advisable if for Mass and Money, Time and Temper- thcode makes provision for indicating this ature. M/s can stand for Mass, M/y can and the reason for non-completion. stand for Money, and T can stand for Time, since in Temperature the Object- ivesare notformally evaluated with A SUGGESTED FORM FOR A RESPONSE test items. SHEET *Thesecondcolumn can contain an A Response Sheet form which was employed abbreviated version of the Objective, or during trialling of these tests, and generally the Key-Words of the statement, so that found to be adequate, is included in this the teacher can recall the particular test book for your consideration. The headings without having to refer back to the list and the marking code were found to be of Objectives when she subsequently quite satisfactory by most of the teachers studies the child's record of responses. who used it during the trialling period. Some schools preferredto use a vertical form *In theCdrdcolumn, the section or sub- uecause it gave space for a greater number section, e.g. "(b) ii", will be inserted. of tests. Some preferred a horizontal form as it gave space for fuller comments. Two There is provision for recording responses to examples of each are placedat the back ten test items; on occasions the teacher will of the bookso that they can be readily perhaps consider five items to be adequate. located. One of each should remain in the Because of the differing numbers of test book for permanent reference; the other, items and the varying number of acceptable markedwithperforationlines, may be responses which may be considered adequate removed for assistance in drafting a stencil- for success in a particular test (ten, eight, master,orforreference whenrekding five,four, or one) itis advisable to note these notes. success or failure for each test (or successes onlyorfailuresonly)inthe column At the top of the page, on the left-hand provided.Anyothercomments about side,thereisa space for details of the performance, such as slowness of working, child being tested; on the right-hand side intermediateI.welreached, distractibility, the suggested marking code is set out. or influence of fatigue, may be entered in the final column if desired. The main part of the sheet has columns for identifying the Objective, for recording the There is NO provision for noting the total requisite number of responses, for indicating numberoftestitems a child answers success offailure,and forwriting any acceptably, or the total number of Objectives relevant comments if and when these are attained. Such total scores would give no considered necessary. guidance as to a child's level of understanding and are meaningless for purposes of The first three columns are for identifying comparison. the Objective being tested. The column for noting the child's answers *Thefirstcolumn is for the number of will be filledin in accordance with the the Objective. The name of the Strand recommended code. This consists mainly of may bewrittenfirst,e.g. Length, ticksandcrosses,butthereareother Number, Time. Then the number of symbols for use on the occasio.s when these the particular Objective (19,20,21 etc.) are not adequate or do not give sufficient will be inserted and will refer to that information. Strand. Alternatively, initial letters may by used before the numerals (as in L2, esA tickwillindicatethat a child V3, Al) to indicate theStrand. Thus N answered correctly or adequately. will stand for Number, S for Shape, L for Length, V for Volume, and so on. A cross will indicate that he answeatd

9 wrongly, inadequately, or not at ail. because of considerations of time, or becauseachildisanswering with A dotwillindicatethatalthough confidence, speed and accuracy over furthertest items could have been a number of Objectives. In such cases presented, the teacher did not present ahorizontallineshould be drawn them because she considered that to acrosscolumns6-10,whichwill do so would have been futile. The dot signify that the giving of these items will signify that there is no intention was considered unnecessary. togivethetestitem, whereas an entirely blank space will mean that If boxes are filled in, this will indicate testing did not take place, or has not that further test items are not provid- yet taken place. ed, or are not possible.

Sometimes it may be decided to give A brief example of the filling in of part of a 0five items only, and not ten, perhaps Response Sheet is set out on the next page.

9 4

10 FILING IN A RESPONSESHEET

11111111

CHILD'S RESPONSE SHEETpage Maii(ing code: 1.1M

acceptable

failed Name: 4;1,Z Class' 1 not presented (futile) Date not presented (urinecessary) 41 d,o.b. /4 / 7/ further items not possible

Objective Objective Procedure Response to test items Success or Comments on performance No. Keywords No. z failure I 3 4 5 6 7 8 9 10 (if required)

N.I9 444 094 00 i r r

I- ZO r r rx rx rx xfad 414444 4 41444414!3 Ar.cl

Bodo1-10 ii Iii Autaseicui 1-20 II cirrl, avA 4440

i 20 letandlouttt APPIr7A Ai, A,4_ "*"14L / . ii AI§/AV Cortigo x.1614 a i r rlir ii r r IIII xAmax+

ilmitmitartio cia104 b r r rIIx x rx itcamt ti:+d, ii 111 I LterigiotI.lo 0+414i 2.ofdie4l i II 11

Example'of suggestedResponse Sheet partially filled in INTERPRETATION OF MARKING reveal Mrs. Jones' selection of keywords to remind her of the Objectives tested. She Itwill be noted that John Smith is aged has decided that for these Objectives, ten about six and a half and that the test took correct out of ten will be the criterion for place about mid-year. John is probably a success, and for quick reference has noted newcomer andMrs. Jones has probably success or failure in the appropriate column. tested him because he does not seem to be For competentresponses with numbers at quite the same level as the other children 1-10 she has however made an exception in in the class, which is beginning work with the number of test items. John was so quick, addition and subtraction and with recording correct and confident in his actions and his number facts discovered. answers that she deemed that five out of five gave sufficient indi ation of success. After three tests Mrs. Jones is fairly certain For Rods 1-20, and for understanding of of John'sstandard. He can wcrk with addition, John was unable to proceed; he discretematerialandwithrodsusing did not know what to do.Failure was numbers to ten; he is quite confident with diagnosed without presenting the full number this.With environmental objects he has of items; dots were inserted to indicate the "picked up" alittleinformation about futility of continuing with further items. numbers to twenty but appears to have had The shading or cross-hatching shown for no formal lessons about them. He is not Objective20 indicatesto any one who confident; although he answered six ques- subsequentlystudiesJohn'ssheetthat tions adequately, he was not himself sure John's one answer constitutes success as that the answers were right. Mrs. Jones does variations of this question are not possible. not consider that he is successful here. With rods there is no ambiguity of interpretation; In the Comments column, it will be noted he has no idea how to proceed; it is obvious that Mrs. Jones has inserted entries both he has had no lessons on rod work beyond about John's test performance and about his ten. John knows the numerals for cardinal behaviour and attitudes. She decided that it numbers and has no difficulties with nought is not necessary to write comments unless or zero. With discrete objects and with rods, they areabout something she wishes to John can combine sets but he has very little remember about achildat the time of realisation of this as representing the opera- testing. tion of addition.

Since John does not yet understand the FOLLOW UP addition operation, it is useless "slotting in" here the problem work about addition, or No suggestions can be offered herefor the use of memorised addition number facts. detailedfollow-upwhendifficultiesare diagnosed. Thesewill vary according to Now that Mrs. Jones has ascertained John's the age of the child, but it is obvious that level of achievement and understanding she teaching or re-teaching, with planned learn- knows the stageat which teaching must ingexperiences,willbenecessary, and continue. possibly further time spent at one particular stage before proceeding to the next stage. A study of the filling in of the sheet will

9 G 12 NOTES ANDCOMMENTS ADVANTAGES OF TESTING which had a bearing on the experiences, in a way they were reluctant to do, or If through testing a teacher can diagnose unable to do, in the ordinary classroom the cause of children's failures then it will situation, revealing facets of personality be possible to plan programmes of help. or thought-processes previously unknown This initself willprobably save further or unsuspected by the teaclinr. failure and make future teaching easier. These arethe obvious advantages of a testing Other information valuable to the teacher programme. was also revealed. Some children performed muchbetterintheone-to-one But there are other advantages, perhaps situation than they did in the classroom. less obvious but equally valuable. They got work rightthat they never could when working in the class situation, Children generally enjoy the experience. but when they returned to the classroom The one-to-one contact with the teacher, after success in the test experiences they the feeling of being special, and being the again performed badly. With such a child centre of attention, often has the effect the problem probably lies not in his of improvingthechild'sattitudeto mental ability or mathematical compre- school, to his teacher, and even to the hension but in his personality perhaps particular topic of testing. heis very distractible,or very easily intimidated; perhaps hehassolittle Sometimes when a child performs well confidence in himself that he changes his in the individual testing situation, there is answers to match those of his neighbours. a carry-over of success when the child Knowing where the problem lies, the returns to work in the classroom situation. teacher can give special attention to his A child who has become discouraged or classroom placement and the influence apathetic, duetorepeatedclassroom of other children on him. failures,may bestimulatedtonew interest and effort in classwork because Sometimestheopposite can happen. of the glow of success which resulted Therewerechildren who performed fromtheindividualworkwithhis well in class, but when they were tested teacher. in one-to-one situations they gave very poor responses. Here the teacher was Often a child who was previously reluc- alerted to a different problem that of tant to talk,to communicate with his the child who is expert at hiding his own teacher, or to enter into discussions with inability and who imitates or copies the classgroups, becomes soenthusiastic responses of others. The child may not aboutthetestingactivitiesthatthe necessarily be cheating, he may behave conversation begun in the testing situation as he does out of laziness or insecurity continues thereafter and the communica- rather than dishonesty. But whatever the tion problem also is overcome. reason, the teacher will realise that there is a behaviour problem instead of, or as Intrialtesting, teachers working with wellas,a mathematical problem and children from their own classes found will organise her teaching and her class that in each case they began to under- management *o deal with this. stand the whole child much better, not justhis mathematical troubles. As the A frequent advantage from testing was children spoke about the activities in the that teachers were alerted to thefact tests they related background incidents that children's failures were often the

139 #4. resultof short-comingsintheir own subsequent probings. But chiefly they are programmes. Many times, vital steps in useless when trying to explainto an learningsequences had been omitted. anxious or ambitious pan. nt, or a busy Sometimes vocabularyhadnotbeen supervisor, that a child is not ready for enriched and extendedsufficiently. the next stage, and will not yet be able to Frequentlytherewasapaucityof cope.Thetangibleevidenceofthe practical experience with real objects and response sheet may furnish the necessary concrete materials. The children had not proof, so that pressures to place in, or handled pyramids, cones, cubes. blocks, promote to, a grade or class where the rods, beam-balances, trundle-witels, tape levels of work are unsuitable or beyond measures, attribute blocks, water, sand, his comprehension may be resisted. and thelike. They had not measured distances,pushedheavyobjects,or Since the Procedures are carried out in a seriated sets of counters. Teachers who one-to-one situation and actually involve realisedthese short-comings were able the child in practical activities as a partici- to rectify the deficiencies by more careful pant and not a mere observer, the need programming, by taking steps to expand forvariedandfunctionalequipment the child's vocabulary, and by including becomes apparent. A good quality beam many more practical activities. balance that works, numerous durable models of solid shapes, varied measuring Teachers who carried out the trialling of tools and the likearenecessities for the tests found that they often received effective testing. And if they are needed unexpected help from thetesting pro- fortesting, how much more are they cedures. A child would failin the first needed for teaching and learning! So the w repetitions of the test items. The testing experiences may alert those in teacher would realisethatshe would charge as to the kind of equipment they need to plan remedial activities for the need to purchase or organise for their child. However she would continue with classesso thatthechildren will have the rest of the test items, commenting agoodfoundationof mathematical on the materials, the language or the experience. actions as suggested in the descriptions, and would find that before the test was Finally, the tests may enable teachers to completedthechild was showing by find outaboutchildren'sattitudes. correct responses that he now understood Attitudes are often as important as know- the questions and would be successful ledge. In one-to-one relationships teachers with the particular Objective. The actual find that they receive a great deal of test procedure became at the same time a information about children's attitudes teaching procedure. Though the child was their attitudes to school, to mathematics, not marked as successful at the time, to experiences, to practical work, to subsequent testing a few days later con- values. An awareness of the children's firmed this finding. Even when this does attitudes is of great assistarre to a teacher not happen, the test procedures at least in knowing how to influence, plan -and indicate an initial approach to remedial organise for improvement. activities.

Thetestresponse-sheetscanprovide tangible evidence of a child's stage of THE TIME TAKEN IN TESTING development, level of maturity, or degree ofunderstanding.They canprovide It will be admitted readily by anyone who is proof of lack in skills, or in knowledge involved with this work that testing is a or in competence. Teachers frequently time-consuming procedure. are heard to assert that they know their classes so well that they can assess these Time is needed for all the aspects of testing. things without any tests. This may indeed oftenbeso.However suchintuitive * Children who can benefit from testing assessments are not always borne out by must be selected.

14 * Response sheets must be made ready. If it is any one of these things, extra sheets of "practice sums" aren't going to do any A place to carry out testing must be good, and the teacher's refusal to diagnose prepared and kept available. the child's difficulty is in effect a refusal to help the child overcome his difficulty. * A "slut" or "slots" in the day's time- table must be arranged and adhered to. The nature of the help needed will vary according to the problem disclosed by the * The most time-consuming of all will testing. In the long run, such diagnosis will be the actual testing of the child, even saveteaching time andpreventfurther though procedures for only four or problems from developing later. five Objectives may need to be given. Furthermore,childrenselectedfor It is hoped that there are not many teachers testing are frequently shy, uncertain, who will be too short-sighted to spare time uncommunicative, hesitant and slow- to helpa child by first ascertaining the moving, which further lengthens the cause of his difficulty. time needed. fhere are teachers who, foreseeing all this, COMMON CA USES OF FAILURE Jecideinadvance nottouse thetests, iecause they take too much time. These Low general ability, and gaps in a child's )eople are saying in effect thatisis too information due to absences from school lime-consumingtohelpachild whose of lack of teaching, are of course frequent lifficulty needs to be diagnosed. and obvious causes of failure. But teachers who carried out tests in the trialling stages t is a teacher's duty to assist a child who is found that two other very common causes 'ailing; itis only sensible to find out the offailure were lack of familiarity with ;ause of the failure first, so that the right relevant vocabulary and lack of appropriate rind of help can be given. experience with materials.

requently help is given which is no help These teachers felt that many of the children itall, because of this failure in diagnosis. would have succeeded had they known the 2onsider the case of a backward nine -year- words being used about the test materials 1d who is continually failing in his "two- in certain of the test questions. Most of ine add-ups". His teacher without thought these words were common words employed lecides that the child fails because he needs ina mathematical context, which showed 'more practice". So he is given more pages that the classroom language must have been, if vertical "two-line add-ups"-- which he in many cases, very narrow, stereotyped ;ontinues to get wrong. This is no help. It and unvaried. "Ordinary" words well within foes not save time. the comprehension of the children were involved(e.g. combine, separate, surface) tis necessary first to consider why the child aswellas morespecialised words (e.g. s failin4. In which aspect of work in addi- pyramid, circular,estimate); these words ion does he lack understanding? wouldhavecausednocomprehension difficulties had they been introduced and tcould be that he hasn't memorized any frequently used as part of the classroom lumber com;s'.nations in addition facts; it practice, routines and activities. ould be that he hasn't memorized some Ifthese combinations.Itcould be the Triallingteachersfeltalsothat children ertical form that worries him; maybe he would have succeeded in many other tests oesn't understand the "carrying" process; hadthey hadaproper background of lerhaps he doesn't comprehend the addition appropriate experiences with actual material Irocess at all; perhaps he can't combine sets and concrete objects on which to base their nd count the new total. Perhaps, even, he conclusions. Children who were able to sort oesn't understand the notions of "more" planeshapes, suchassquares,triangles, nd "less". circles, ,were quite unable to deal

15 9 Q IJ with spheres, cones, cubes, ,although table is, but should not be expected to find solid shapes are not inherently more difficult itsareaaccurately by coveringitwith to work with than plane shapes in fact informal units, although of course he may be they are probably easier. It seems a logical given initial experience in the covering of learning sequence to proceed from solids to a surface. surfaces,thenplaneshapes, boundaries, lines (segments), and finally points. It would It should be noted that the tests necessary appear to be sensible to provide lots of for kindergarten childre,1 in any strand are wooden, plasticor cardboard models of very few, as these children are still in the pyramids, cones, cylinders, prisms and so early experiential stages. on for children to play with, talk about, build with, stack up and pack together The num4 c, of tests suitable for YearI some sets of each, similar in size, other sets childrenwillbeslightlygreaterasthe in varying sizes. In this way children would teacherseeks todiscover if kindergarten learnthe names and the properties with concepts have been achieved. verylittleeffort,andthentheirlater explorations would be much more meaning- By Year 2 the need for diagnosis will be ful. Clock-faces, balances, metre-sticks and greater still as the child has had more work other measuring devices were strangeto presented to him wherein he may have many children and so they performed badly failed, and so the range of possible tests will with them. The cause of failure was not be wider. lack of understanding on the part of the children, but lack of experience, both in The full number of tests will probably be of using the material, and in talking about it most use with Year 3 and Year 4 children with adequate vocabulary. who are failing in primary work because for some reason they are not competent in, or Most of thetriallingteachersfeltthat conversant with, the work normally treated children could be introduced to appropriate in Infants' grades. (Hence the inclusion of mathematical vocabulary at a much earlier the phrase "K-2" in the title of this book.) stage than many of them are at present, and that children would not only cope with it easily but would enjoy and benefit from SUGGESTED GRADE-AGE LEVELS FOR the activities in which they discussed the DIAGNOSTIC TESTING materials they were using by means of a wider range of mathematical words. The tests are designed so that teachers can find out whether a child is having mathematical difficulties because of a gap (or several TESTING BEYOND GRADE LEVEL gaps),in his knowledge, or his skills, or his understandings. A gap of this description Thesetests are not designed to discover may have been caused because there was no giftedchildren,nortoextend"able" previous teaching of the subject-matter in children. question,or because thechild did not assimilate or profit by previous teaching if Itis not intended that in normal circum- itdid take place. As soon as the child's stances a child should be subjected to testing present teacher has identified where these for Objectives relating to topics, skills or gaps are, she may provide help by giving the understandings which are beyond his grade missing lessons, and planning for practice level. For diagnostic purposes or to ascertain and consolidation of their content. Such whether teaching has been successful, this is diagnosis is particularly necessary for eight- both futile and unnecessary. and nine-year-old children in primary grades who are failing in work normally learnt in In Volume, forexample, a kindergarten infants' grades. However, the tests are equal- child might be expected to know about full lyusefulfor infants' children who have and empty, but should not be questiond failed to learn the work of a previous grade. about the need for a standard unit; similarly he should know what the surface of the For this purpose certain tests are appropriate 3 06 foruseinthe various grades. Suggested children aged five years, in theirfirst Grade-Age Levels are set out below. year at school.

Year 4 in these listswill indicate, in When a childis tested for work normally general, children aged nine years, in their given in a previous grade, the teacher can fifth year at school. ascertain if it has keen assimilated. Year 3 will indicate, in general, children When a child is tested for work given in aged e^ght years, in their fourth year at his own grade, the teacher can ascertain school. the child's present level of achievement, or his present stage of development (i.e. Year 2 will indicate, in general, children where he "is at"). aged seven years in their third year at school. In the text, the tests are set out in continuous form without age or grade labels, for Year I will indicate, in general, children use with individual children, for the purpose aged six years, in their second year at of monitoring their progress through the school. various stages toa level appropriate for each child. Kindergartenwillindicate,ingeneral,

SUGGESTED LEVELS

Year 4 Year 3 Year 2

Tests suitable for Tests suitable for Tess suitable for administra- administration, if administration, if t ton, if required. required. required. (The time of the year when the test is given will influence Number: All tests. Number: All tests. theselection of Objectives Other Strands: All tests. Other Strands: All tests. and the choice of levels.)

Number: All tests, by end of year. The more difficult aspects of work with fractions and division may be omitted if desired.

OtherStrands:Alltests, by end of year.

17 SUGGESTED LEVELS (continued)

Year 1 Kindergarten

Testssuitableforadministration,if Testssuitablef'nAminist ration,if required. required. (The time of the year when the test is (The time of th...-a, when the test is givenwillinfluencetheselectionof givenwillinfluencetheselectionof Objectives and the choice of levels.) Objectives and the choice of levels.) Number: Number: Objectives Objectives

1- 29, 1 - 17, 34, 19 (a) i, 37 (at an appropriate level), 20 (i), 38 - 40 (all at appropriate levels). 21 (a) i.

Shape: Shape: Objectives Objectives

1 (at an appropriate level of vocabulary), 1 (at an appropriate level), 2, 2 (at an appropriate level), 3 (at an appropriate level), 3 (at an appropriati 5 (at an appropriate level), 6 (vocabulary at an appropriate level). Length: Objectives Length: 3 (not waste paper tin), Objectives 10 (at an appropriate level), 1,3,4, 12 (vocabulary at an appropriate level). 10 (at an appropriate level), I1 (at an 'propriate level), Mass: 12 (vocabulary at an appropriate level). Objectives 1, Mass: Objectives 10 (at an appropriate level), 1, 2, 3, 4, 12 (vocabulary at an appropriate level). 5 (two objects only); 6, 10, Volume (Capacity): 11 (at an appropriate level), Objectives 12 (vocabulary at an appropriate level). 1, Volume (Capacity): Objectives 5, 10 (at an appropriate level), 1, 11 (vocabulary at an appropriate level). 2, 3 (Procedures i and ii), 5, Area: 6 (at an appropriate level), Objectives I 0 (at an appropriate level), 1, 11 (vocabulary at an appropriate level). 2 (at an appropriate level).

Area: Time: Objectives Objectives 1, 1, 2 (covering surfaces only, for a few 2 (o'clock only), children only), 3 (week days), 4 (at an appropriate level), 6 (vocabulary at an appropriate level). 5 (Procedure 1, for some children).

3 9 18 SUGGESTED LEVELS (continued)

Year 1 Kinergartend Cont. Cont. Time: Money: Objectives Objectives 1, I (not 50 cents), 2 (o'clock and half-past), 6 (one dollar note). 3 (at an appropriate level), 6 (at an appropriate level). Temperature: Objectives Money: I and 2 (both at appropriate vocabulary Objectives levels). 1, 2, 3 (for some children), 4 (for some children), 6.

Temperature: Objectives 1 and 2 (both at appropriate levels).

MEMORISATION and PROBLEM WORK These tests are intentionally detailed and AND GRAPHS thereforelengthy.Itisof course quite possible to design shorter tests composed of Although the Objectives have, in general, fewer examples. But it must be realised that been writtenina logical and sequential the tests in this book are presented not as order it is not possible to maintain a strict tests of knowledge or achievement, but as teaching/learning sequence throughout. diagnosticprocedurestodiscoverthe particular area or level of difficulty in the Two particularfacets must be regarded memorisation or use of number facts. separately as to their place in the sequence. These are the testing of memorisation of (b) Testing competence in problem work number facts, and the testing of competence and graphs. in problem work and graphs. Testing of knowledge of the different (a) Testingmemorisationanduseof aspects of problem work and graphs is Number facts treatedin Objectives 38, 39 and 40. But itis not intended that .testing of All the memorisation tests are outlined theseshouldbedelayeduntilthe in Objective 37 but it is intended that previous 37 Objectives have been tested. teachers insert the relevant parts of the Testing in Objectives 38, 39 and 40 test at appropriate places in each child's should be slotted into the appropriate learningsequence.Forthisreason places following study of the relevant categories are specified throughout the operations and numbers. tests for memorisation so that different levels can be discerned. This is not only to make it possible to pin-point areas of USE OF "CRUTCH" FIGURES difficulty for an older child. Itis also so that teachers can select the proper When working withthe more complex categories for younger children. types of number sentences, such as

19 33 4 + 2 + 2 = El and 9 4 2 = E many If a kindergarten child has a good idea of childrenmakeuseof "crutch"figures the word "surface" and of the surfaces of (small numerals representing partial answers) things around him, he will later have a writtenabovetherelevantpartof the better chance of understanding Area than a sentence. The numeral 6 may be written child who is merely given a fomiula (e.g. above 4 + 2, and the numeral 5 may be "1 x b") at the age of nine or ten. written above 9 4in the examples above. Earlyexperiencesinmeasuring witha Number sentences completed correctly using variety of units, in making simple graphs, this device may be accepted on an equal in playing shops with money, in handling basis with those completed correctly with- models of cubes, pyramids, cones and other out it. solids,all provide interestas greatas in Number, but at the same time are designed Some teachers may prefer to see use of this to lead to acquisition of necessary concepts, device as indicating an intermediate stage, to suitable for the child's stage of development, be accepted, and commented upon as such which willfacilitatefuture learning. Itis when recording the child's progress. The not fair to deprive him of these experiences childwill then be expected to complete merelytoconcentrateonthe Number similar number sentences atalater stage strand. without use of "crutch" figures, showing that he can mentally retain sub-totals for Iftheseideallearning-times are missed, a longer time than previously. subsequent learning is often more difficult. It should be noted that the use of "crutch" figures by children having difficulty gives the testerincreased opportunity for CONSERVATION diagnosis, and for these children at least the useof "crutch"figuresshouldnot be Although an understanding of Conservation discou raged. isvital to successful work, the concept is developed onlyafteralarge number of contributory experiences. For this reason, testing about Conservation is set down near "NUMBER" BIAS the end of thelist of Objectives for the appropriatestrandandnotfirst.The Concept formation beginsinthe earliest development of the concept is slow, and is years.Thisissoinallthe strands of not uniforni throughout all the strands. Mathematics whicharesetoutinthe curriculum, and not only in the Number Forteachers who wishtofollowthe strand.Trialling experiences suggestthat development of this conceptin children many of the tests in Area, Shape,Volume, the relevant tests are set out below. It is not Length, Mass, Money, Time, and/or Temp- expected thatthe concept willbe fully erature could not he given because there had developed throughout, atleastuntilthe been no teaching of the topics covered in age of seven years. Five- and six-year olds the test items. Sometimes this was because may be given thetests, and their results the topics had inadvertently been omitted will show a variety of intermediate stages. from the class programme. However, it was No consternation need be felt about this; found on many occasions that the teacher all that is indicated is the need for further had consciously decided to unlit these other experiences, and an understanding of the strands as unimportant and to concentrate child's present stage of development. on number concepts and experience. Procedures appropriate for ascertaining the This is unfair to children. The early concepts stage of development of the Conservation in the other strands are just as important conceptinparticular strands are set out tolater development inthose strands as below: early numberconcepts aretolater arithmetic studies. Number: Objectives 6, 15. 16, 22.

3.4 20 Shape: Objective 2 (ii) may have a bear- diagnosis of responses when the wording ing on the development of the was altered in statements and questions to concept. children. Those teachers wondered whether the changes they made would constitute Length: Objective 10. "prompting", even thoughtheywere informed thatit was permissible to make Mass: Objective 10. changes in the interests of communication with the child. Volume (Capacity):Objective 6. The following suggestions are given. Area: Objective 5. * Where the changes are mere alternative Money: Objective 2 (The different ways forms in the choice of words or phrases, of making up the same amount and are substituted to accord with the may have abearing onthe teacher's usage familiar to the child, this development of the concept.) will have no effect on the assessment of the response.

GENERAL KNOWLEDGE * Whereradical changes must be made because the child has no knowledge of Questionsregardingapplicationsinthe the vocabulary in question, this can affect environment (Why are plates round? Why the result; the tester may be unable to are triangle-shapes used in bridges? Why the discover if the childeally understands diagonal on wooden gates?) are not intended and could be successful in a particular to be questions on General Knowledge. They Objective. She may feelthat he would are designedto ascertain how much the be capable of su:cess in spite of the child has understood and retained from class necessity to explain the words: if so she lessons and discussions about these matters, should note this on his Response Sheet, or in the case of a newcomer whether the but should also note his failure with the childhas had suchteaching andifso, correctvocabulary(andinfollow-up whether he has understood it. should includethenecessary language work!). Instead of individual questioning ina test situation the teacher may prefer to base her *Itisnot permissible to make changes assessment of the child in a class situation which alter the nature of the question or during discussion of these topics, and then give clues as to the answer required and on his Response Sheet she will note perhaps still mark the child as correct. If such the thoughtful relevant nature of his contri- changes are made an intermediate stage of butionstothediscussion and how well understanding may be discovered, which they indicate his grasp of the ideas. should be recorded on the child's sheet. After further experiences using adequate Similarly withtestitems concerning the vocabulary, the child may succeed fully digital clock, the trundle wheel, the calendar, when retested. andthethermometer, the questions are designed to follow class experiences, or to * Whenever an alterationto wording is indicate the lack of class experiences. They made which thetesterfeels may con- are not designed to probe the home back- stitute prompting, or have a bearing on ground or discover any general knowledge the child's response, this should be notee. otherwise acquired (useful though this may in the Comments column of the Response be). Sheet.

PROMPTING The frequent occurrence of this problem indicated the "narrow" nature of the vocab- Some of theteachers who carried out ulary employed, and the limited use of preliminary testing were concerned about alternatives, in some classrooms.

21 dei Words such ascombine, separate, pattern, suggested that four correct responses out subtract,represent,difftrence,compare, of five can be taken as conclusive in most match, remove, pair, equalandcomplete cases, but that sometimes a score of eight or could be used much more often in context nine out of ten is more reliable. and would broaden the child's experience and communication, aswell asmaking This is because the four correct responses assessment of mathematical understandings may be the limit of the child's knowledge more accurate. and he could get no more answers right no matter how many further questions were asked. Four out of five would become four WRONG RESPONSES out of ten, which could not be counted a successful effort. It is important not to tell a child he is wrong in his answer, and for the tester to receive Further, a child may answer correctly after all responses with equal interest. A child's a great deal of thought, hesitation and self- subsequent responses can be influenced by correction, and may not appear confident the tester's reactions to answers. Similarly in his approach. The tester may then not itis important not to let the child observe be sure whether four such responses were the tester's crosses and ticks while responses rightafter much effort, or were purely are recorded.Children sometimes are "lucky" guesses. Eight such correct responses diverted by trying to work out the code would earn the child the right to be labelled and do not then attend properly to the next successful, although a note in the Comments question. column about his performance would be desirable (and perhaps re-testing at a later It has been noted that one correct response date to see if he is then more confident in is not, generally, sufficient to assume success. the situation). In the same way, the tester must not assume that if the first response is incorrect, failure Also, many young children frequently waver is to be recorded for the particular Objective. for a long time betweenknowingandnot If a child confirm,: to give answers for the knowing.This can happen in many learning test items then probablyfourwrong situations, e.g, in writing b and d correctly, responses would justifythedecision to in knowing the right and the left hand, in record failure, and to discontinue testing. spelling a difficult word. The child may be Failure must be recorded earlier than this right to-day, wrong to-morrow, right again incases where a child makes one wrong the next day. This happens too in mathemat- response and then is unwilling, or refuses, ical matters,'and is another reason for being to try further. cautious about accepting a single response as conclusive. When itis felt that a child is Sometimesitisunwise to continue with at such a stage in the development of a skill questions when the child knowsallhis or an understanding, it may be advisable to answers are incorrect; this often discourages give the test over several days to be sure of or upsets the child, and can influence his the child's consistency. attitude to the next set of test items. The tester must rely on commonsense in the case In the majority of cases the tests are short, of wrong responses, just as much as in the and the test questions are simple; itis not case of correct responses. much more difficult to give five items than to give one. Similarly itis not much more time-consuming to give ten items than to HOW MANY TIMES? give five. The child will not become bored or fatigued if the tests are presented in an The question of how many times a child has interesting manner by the tester. to respond correctly for success must be considered in relation to each Objective. Nevertheless there will be other cases where five or ten test items will not be appropriate. Intheinstructionsfor administering the tests andmarking them,ithas been How many times must a child count to a

11 3 6 hundred to prove that he can count to a Let us say at once you don't have to read hundred?If he counts quickly and con- all the tests before you can use the book; fidently, without hesitation, passing from the introductory pages will explain how 39 to 40, 59 to 60, and so on, without the tests are to be given and used. faltering, then surely he doesn't need to repeat the performance further to convince Let us ALSO say at once that it is not you certainly not five times, let alone intended that anybody will need to use ten! all the tests. Four or five tests per child- in any particular strand will probably be Counting to a hundred is not one of the tests quite sufficient for a diagnosis. in the book, but in cases such as that, where the test is lengthy, and the child perfectly The sizeof the book is brought about confident and perfectly accurate, there is because the Number, Shape and Measure- no need for further repetitions. ment strands are all included. It was con- sideredatfirstthat three separate books Gi Lel cases where repeated test items are might be issued; teachers when consulted unnecessary may be suggested. Where the indicated that they nearly all preferred the tester is giving preliminary tests to ensure inclusion of all tests in one volume. that the child is comfortable in the situation, thereis no need to give repeated items. The amount of the content is brought about Where preliminary tests are being given to because the range of tests must be compre- as a suitable starting place there is hensive. A child will have difficulty with to give more than one or two items. only a few aspects of his work to date, and usually it will only take five or so tests to To p,:rsist with numerous repetitions in such ascertain which these aspect.- are; however circumstances will cause both tester and they will differ from child to child. Some child to 67'144 extreme boredom and fatigue, childrenwillhavedifficultywitheay nottO 7,0fitiondistasteforthe whole concepts, some with later skills. Some will programme! have difficulty in memorizing certain facts, some will not understand certain types of Since the tests are not intended to be stand- environmental situations when expressed in ardised procedures, the number of repeti- words. Nobody can know in advance which tions does not have to be uniform. It is left tests are going to be needed, hence all are to the tester's experience and corrunonsense included here. An all-inclusive volume of todecidewhicharethetestswhere this nature is necessarily lengthy. Guidance repetitionsare necessary, and how many is given in the Introduction and the Index repetitions should be given. Where itis of Objectives as to how the tests may be decided not to give a number of test items used. The comprehensive nature of the book this will be because of the complicated or causesitssize and thickness but thisis lengthy nature of the tests and the child's intended to be an advantage to teachers not competence in responding. (But if the child a hindrance; the range of tests is d == signed isnot confident, andhis responses are to increase the range of assistantwhich inconclusive, repetitions will be needed even teachers can give to children who may be though the test is lengthy.) having difficulty.

SIZE OF THE BOOK WHY "SETS" AND "SUBSETS"?

The thickness of this volume, and the If mathematics is going to mean anything to comprehensive nature of the contents, have young children it has to be real. They should proved daunting, not to say "off-putting". discover it, apply it and understand it in to many teachers at first sight. They think terms of their own environment and the thatit will not be possible to read itall things in that environment. When they have before using it. They think also that they will had experience with real things they may not be able to administer all the tests in it supplement their work by using mathemat- and therefore they will not have a use for it. icallystructured materials, particularly in

23 ,3 order to organise and tabulate number facts. alternative formsofdescription, for But if they do not und,.Jstand number facts ex ample, "this collection of things", "some in terms of the real objects in their environ- of these things", "another lot of things", ment we cannot be sure that they under- "therestof the things","this group", standNumberat all.Hence intesting "these smaller groups", and so on. Only one mastery of the Number Objectives whenever ofthewords used ("subset")isatall possibletestshavebeenwrittenusing specialisedandnotfoundinordinary objects which are familiar to the child in his conversation. If desired, this word can be home and his schoolroom. avoidedandsubstitutionsmade;most triallers found no difficulty and preferred These tests are described for the teacher's its brevity to the possible alternative circum- benefit using words such as set, member, locutions. The emphasis in these tests is not element, combine, partition, separate, and on the words but on the activities they so on, and sometimes in the questions and descrii-P and the ideas they convey. requests to children some of these words are also used. Itis stressed that there is no intention of introducing ortesting knowledge of Set Itis pointed out to readers and testers that Theory or the correct use of its vocabulary. these words are used in explanation of the tests about environmental objects. They are The words are used in an environmental used with the same sense and meaning as contexttoprovide clear descriptions of they would have in ordinary conversation. activities,widenandenrichvocabulary, They providea much more concise and relate mathematics to every-day life, and exact form ofcommunication than enable children to express ideas concisely.

24 THE TESTS

39 NUMBER

OBJECTIVES. TEST MATERIALS TEST PROCEDURES and COMMENTS.

The Child.

1 Identifies The child's classroom PROCEDURE: attributes environment.

(using the Say: "Show me something blue," language of "Show me something rough," attributes) "Show me something round," "Show me something wooden," "Show me something io write with." "Show me something heavy,"

Make five or ten such requests to give the required number oftest items. The attributes given above are suggestions only. The tester should select those attributes considered the most appropriate for future development,

Acceptable Response:

The child points to indicate the object with the appropriate attribute,

2.Classifies an i. About twenty objects i. PROCEDURE: object by different in colour,

means of its shape, size, usage, Place all the objects before the child. attribute(s) composition and/or texture. Say: "Show me a rough block," "Show me a blue bead," "Show me a round thing," "Show me a wooden box." "Show me something to write with," "Show me a heavy toy." "Show me a big red tin,"

Continued NUMBER 0.=ian....n. OBJECTIVES, TEST MATERIALS TEST PROCEDURES and COMMENTS,

The Child,

Make five or ten such requests to give the required number of test items,

Acceptable Response:

The child indicates the correct object by speakingor pointing.

Notes: Suggested teacher inst'uctions are basedon each of the following categories:shape, colour, size, usage, composition, texture, and conjoint attributes (i.e. two together).

If a child experiences difficulty withany of the categories, the teacher should retest with further questions concerning that category,

ii. About twenty objects ii. PROCEDURE: different in colour,

shape, size, usage, Show the child five, or ten, of the objects one ata time, For each object, say: composition and/or texture. "Tell me something about this."

Acceptable Response:

The child makes responses in the form:

"It's red." "It's made of plastic." "It's heavy." "It's soft." "You write with it."

Note: Ifachild gives a response of the type "It'sa pencil" (i.e. name only), continue: "Tell me something else about it," NUMBER

OBJECTIVES. TEST MATERIALS TEST PROCEDURES and COMMENTS. The Child...

3,Sorts objects. Collections each of about PROCEDURE: fifteen objects with clearly defined attributes. Place one of the collections of objectsbefore the child.

Collections of objects may Say: "Sort these things for me,"or be based respectivelyon the "Put the things together that belongtogether." following categories:

When the child has carried out thisdirection, say (indicatingone group, then another, colour shape

colour and shape "Why did you put these things together?" environmental usage

ownership Repeat, to give the required number of test items, using other collectionsone at a time.

Acceptable Response:

The child justifies the basis of hissorting,

Note: These test itemscover a range of understanding at thesame level, If, however, the child succeeds with onlysome of the categories, the teachermay design test activities, for each of the categories,to ascertain the particular category causing difficulty. NUMBER

TEST MATERIALS TEST PROCEDURES and COMMENTS.

PROCEDURE:

A selection of i. Place the diverse objects before the child. diverse objects. Say: "Makea set of buttons."

".A selection of toy Place the toy animals before the child. animals, Say: "Makea set of cows." iii. A selection of iii.Place the counters before the child. counters, Say: "Makea set of blue counters,"

iv,A selection of beads. iv,Pike the beads before the child. Say: "Makea set of big round beads."

A selection of attribute v,Place the attribute blocks before the child. blocks. Say: "Makea set of small thick square blocks."

Note: Other suitable Any of the activities may be repeated, lising slightly different nterial, to give the categories of objects may be required number of test items. substituted, and the test instructions adjusted accordingly. Acceptable Response:

The child performs the appropriate action.

Note: There is a range of difficulty among the test activities. Ifa child fails in carrying out any of the test items, the teacher should retest with at least five activities each for i,(generalappearance), ii.(particular appearance),iii. (one attribute),iv. (two conjoint attributes), to ascertain the particular aspect causing difficulty. NUMBER

OBJECTIVES. TEST MATERIALS TEST PROCEDURES and COMMENTS. The Child...

5.(a)Pairs the i, Two non-equivalent \ (q) sets \a)PllOCEDURE; elements ofone of similar objects (e.g. set with the four buttons and six Place the two sets describedin (i) before the child. elements of buttons) Say: "Pair the elements (things) of these sets." another set.

Two equivalent sets of Repeat, using the sets for ii, iii, iv andv; any of the activities may be repeated similar objects, (e.g. five using different objects, to give the requirednumber of test items. Note: The word marbles and five `member" may be marbles) Acceptable Response: lubstituted for

'element" if iii. Two equivalentsets of The child performs theappropriate action. Sometimes all the elements ofone set lesired. dissimilar objects (e.g. will be paired u.iith all the elementsof the other, sometimes there will be elements seven pebbles and of one set which cannot be pairedwith a member of the otherset. seven counters)

The pairs may be placed randomlyor in an ordered fashion, iv. Two non-equivalent

sets of dissimilar e.g. objects, (e.g. fourpegs 00 and six beads) 000 00 0 0 0 0 0 v, Two non-equivalent 0 sets whose members coor are all dissimilar (e.g. a 000111 set of four consisting 00000 of a marble,a bead, a 00 0 block, a counter, and 0 0 a set of five consisting of a bottletop, a Note: If desired, the teachermay test with at least five activities each for i pencil, a thimble,a iii, iv and v. toy car, a shell).

Continued NUMBER

OBJECTIVES, TEST MATERIALS TESTPROCEDURES and COMMENTS.

The Child,

5. (b) Finds whether (b) As for 5(a) iv. (b) PROCEDURE: sets match or do not match by i. Two non-equivalent Place the two sets described in (i) before the child. using the sets of similar objects Say: "Find out whether these sets match or do not match," technique of (e.g. four buttons paring. and six buttons) Repeat, using the sets for ii, iii, iv and v; any of the activities may be repeated using different objects, to give the required number of test items. ii. Two equivalent sets of Note: When sets dissimilar objects, (e.g. match they are seven pebbles and said to be seven counters) Acceptable Response: equivalent. iii. Two equivalent sets of The child correctly performs the action of pairing the elements of one set with dissimilar objects, (e.g, the elements of the other. The pairs may be arranged randomly or in an ordered seven pebbles and fashion. seven counters) If all the elements of one set are paired with elements of the other set, he states that iv. Two non-equivalent the sets match. sets of dissimilar objects, (e.g. four If there are unpaired elements in one of the sets, he states that the sets do not pegs and six beads) match.

v. Two non-equivalent sets whose members are all dissimilar (e.g. a set of four consisting of a marble, a bead, a block, a counter, and a set of five consisting of a bottletop, a pencil, a thimble, a toy car, a shell). NUMBER

OBJECTIVES. TEST MATERIALS TEST PROCEDURES and COMMENTS. The Child.

6. Understands For each activity, two PROCEDURE: that two equivalent sets of related

equivalent sets objects, i, Place two equivalent sets of related objectsbefore the child. Match the setsany remain equiv then ask (for example): alent however For example: the elements "Is there a knife for every fork?" of each set are i. knives, forks re-arranged. Move the knivesa little apart, and repeat the question. ii,cups, saucers

Restore the sets, thenmove the forks closer together and repeat the question. iii.triangles, strikers

Restore the sets, thenmove the knives widely apart and the forksa little apart iv,flowers, vases repeat the question.

v,books, pencils iii,iv, v, Repeat the procedure for all theexamples, varying the order ofre arrangement. Note:As a final test, if desired, the procedure may Any of the activitiesmay be repeated, using other sets of related objects, to give till be repeated using equivalent required number of test items, sets of unrelated objects,

e.g. marbles, crayons. Acceptable Response:

The child succeeds if he maintains that there is equivalence of sets throughal. arrangements,

Note:The more widely the objects in thesets are spaced, the more likely it i! that the child cannot maintain equivalenceof sets. Therefore these experience: need to be repeated constantly until understandingis established. (The concept it conservation) 46 NUMBER ....=.4 OBJECTIVES. TEST MATERIALS TEST PROCEDURES and COMMENTS,

The Child.

7.Counts the Sets of objects, in number PROCEDURE: members ofa within the range 1 -.10. set. Place a set of objects before the child.

Say: "Count the members, inany order." or "Count these things (beads, cars, ) for me." Later, within the range

1.20. Repeat, to give the required number of test items, using other sets, the members varying in number; some sets to have similar, and some to have dissimilar members.

Acceptable Response:

The child succeeds if the counting sequence is correct, andno member is missed of counted twice. The tester should recordhow farthe child can consistently maintair one-to-one correspondence while counting (e,g.to ten).

Note: The child is not required to state the cardinal number of theset.

3, Finds the Sets of objects, in PROCEDURE: cardinal number number within the range of a set. 1.10. Place a set of objects before the child,

Say: "Count the members (things, cars, .). How may in the set?"

Repeat, to give the required number of test items, using other sets, the members varyini in number; some sets to have similar, and some' to have dissimilar members.

(Continued over', NUMBER

OBJECTIVES. TEST MATERIALS TEST PROCEDURES and COMMENTS, The Child.

8. (continued) Acceptable Response:

The child succeeds if he counts correctly,and states, (for example):

"Seven" or "Seven things"or

"There are seven members (things,cars,. ) in the set."

Note: The child may count silently or aloud. If the child counts silently andgives a incorrect answer, ask him to count aloud.

9.Makes a set A collection of objects. PROCEDURE: containing a given number of Place a collection of objects before thechild. elements,

Say: "Make a set of six."

Ask:"How many 'things' in that set?" Say: "Show me one object." Ask: "How may 'ones' in that set?"

Repeat to give the required number oftest items, on each occasion asking fora se containing a different number of members, andquestioning the child as to the numbe of elements ("ones") in the set.

Acceptable Response:

The child makes. the set correctly andindicates by his response that he understands fc example, that a set of six consists of six elements (six. "ones ").

Q, NUMBER ...,..+...... r,,.....+vm*,.. OBJECTIVES. TEST MATERIALS TESTPROCEDURES and COMMENTS. The Child.

10. Matches numeral A set of numeral cards in i. PROCEDURE: with cardinal the range

number, 1-5 Place a set of objects before the child, andgive him the appropriate set of numeral 5.10 cards. 1.10 and, later,

1.20 Say: "Count thisset and find the right (numeral) card."

Sets of objects with Repeat, to give the required number oftest items, using other sets within the similar or dissimilar selected range. members within the

range Acceptable Response: 1. 5

5.10 The child selects the appropriate numeral card. 1.10 and, later,

1.20 Note: For an older child begin with therange 120. If the child finds difficulty at the later levels, give five tests at each of the earlier levels indicated.

ii.A set of numeral cards ii.PROCEDURE: as for (i) and a rubber ball. Bounce the balla number of times, within the numeral card range, and ask the chile to find the appropriate card.

Repeat to give the required number oftest items, making a different number of bounces on each occasion.

Acceptable Response:

The child selects the numeral card which matchesthe number of bounces.

(Continued over)

A') NUMBER 11.11.M.MMI.M1/ 11..MMlin1111.1INIIMM

OBJECTIVES. TEST MATERIALS TEST PROCEDURES and COMMENTS.

The Child,

0. (continued) iii. A set of numeral cards as iii.PROCEDURE: for (1) and a number line drawn or painted on the Place the child in front of the number line. Enclose seven spaces between your chalkboard, without hands, or run your finger over seven spaces, in each case starting from the beginning numerals as shown: of the line.

Say: "What number am I showing?"(seven)

Give the child the appropriate set of numeral cards.

Say: "Show me the numeral card that goes an this dot," (indicating the last dot of Note: There is t at the the number illustrated, i.e. the position for the numeral 7). starting point. Repeat, illustrating a different number on each occasion, to give the required number of test items.

Acceptable Response:

The child selects the appropriate numeral card.

iv.A set of numeral cards as iv.PROCEDURE: for (i) and a box of structured material, e.g. Hold up a numeral card. Cuisenaire rods, (Indicate the unit) Say: "Show me the rod that goes with this card,"

Hold up a rod.

Say: "Show me the numeral card that goes with this rod."

Repeat, using different rods and cards, to give the required number of test items.

Acceptable Response:

The child selects the appropriate rod or numeral card,

11 NUMBER

OBJECTIVES, TEST MATERIALS TEST PROED'iRES and COMMENTS. The Child, ,

11, Writes numeral As in 10 for i, ii, iii, and iv. PROCEDURE: for cardinal

number of a set. Pencil and paper, or As for 10 (with at least five test items each for i, ii, iii and iv)except that, instead of selecting the appropriate numeral card, the child is requestedto write the required Chalk numeral.

Say (for "Write the numeral thatgoes with this set of things." (for "Write the numeral thatgoes with this number of bounces." (for "Write the numeral thatgoes on this dot." (for iv): "Write the numeral thatgoes with this rod,"

Acceptable Response:

The child writes the appropriate numeral,on paper or chalkboard, for each activity.

12. Uses ordinal i. A set of objects i, PROCEDURE: numbers, arranged in a line.

Place the line of objects before the child. Indicate the first object.

Say: "This is the first thing." "Show me the sixth thing." "The last thin ." "The third thing.""The second.""The ninth."

The child points to the appropriate object.

U Then ask:"Which object is this?" (pointingto an object in the line). "And this?""And this?"

Repeat, to give the required number oftest items. (Continued over)

51 NUMBER

TEST MATERIALS I TEST PROCEDURES and COMMENTS,

Acceptable Response:

The childsays, for example: "The fourth one"."The second one," "The ninth one."using the appropriate ordinal words.

H. A number line without PROCEDURE: numerals, as shown: Say: "Look at the number line. Point to the third dot.""The seventh," "The first."

Continue, indicating other dots.

The child points to the appropriate dot.

" Say: "Now look at the number line. Which dotis this?""And this?" "And this?" Note: There isno dot at the starting point. Indicate different dots, to give the required number of test items.

Acceptable Response:

The childsays, for example: "The first dot," "the third,""the fifth," using the appropriate ordinal words. iii, A rubber ball iii. PROCEDURE: (or a percussion

instrument.) Place the ballnear the child.

Say: "Get the ball. Now bounce the ball;count as you go; stop after the sixth b ounce ."

The test proper begins here. Continued NUMBER

OBJECTIVES. TEST MATERIALS TEST PROCEDURES and COMMENTS. The Child

12. (continued) The child carries out the action correctly.

Say: "Watch while I bounce the ball."

** Bounce the ball several times in quick succession.

Say: "Which bounce did itstop at?"

Repeat, making other numbers ofbounces, to give the required number of test items.

AcceptableResponse:

The child uses the correct ordinal word in eachresponse, for example, "the third bounce."

Note: If bouncinga ball presents a difficulty for the child, substitute tapping on a percussion instrument.

**The test proper beginshere. NUMBER

OBJECTIVES. TEST MATERIALS TEST PROCEDURES and COMMENTS.

The Child .

13. Orders sets PROCEDURE: (counting order Containers with sets of one

to ten objects, and, later, Place the containers before the child in random order. one to twenty, placed in

random order, Say: "Put the sets in order fromone to ten." (Later, "from one to twenty.") Numeral cards 1.10 (later 1.20) Rei-i.:, to give the required number of test items, presenting the child with the containers in a different order on each occasion.

Suitable containers would

be: Acceptable Response: saucers

small shallow boxes The child arranges the sets in counting order. Hemay first count to find the carding icecream container lids, number of each set, and label it with the appropriate numeral card.

Note: If the child fails in the test using sets one to ten, repeat usingsets one to five If, when commencing the test using setsone to twenty, the child fails to place then correctly repeat the test using sets one to ten. NUMBER

OBJECTIVES. TEST MATERIALS TEST PROCEDURES and COMMENTS. The Child

14. Understands the i. A number line with PROCEDURE: relationship of numerals to 10 and, one set to the later, to 20. Place befure the childa number line with numerals. next set in

counting order. Say: "Show me the number eight." Ask: "Can you showme the number nine? How many more is that?" 1 2 34 5 67 89 10 or Note: There is no dot at "How manymore is nine that eight?" the beginning of the line.

Repeat, using other pairs of consecutive numbers,to give the required number of test items.

Acceptable Response:

The child replies, without counting: "One more."

ii.Containers with sets of ii.PROCEDURE: one to ten objects and, Place the containers in counting order beforethe child. later, one to twenty, placed in counting order. Say: "Point to the sixth container. Nowpoint to the seventh. Howmany more Each container is to be in that one?" (Indicate the seventh,) marked with the appropriate numeral. Repeat, indicating other pairs of adjacentcontainers, to give the required number of test items.

Acceptable Response:

The child replies without counting the members ofthe sets: "One more."

Note: If the child wishes to count the members ofthe sets before responding, he may do so. This however indicates an intermediate stage of understanding. NUMBER ,rmw.,..,:m111101111.M.M110

OBJECTIVES. TEST MATERIALS TESTPROCEDURES and COMMENTS.' The Child1 ..

[5, Understands that Sets of objects within the PROCEDURE: a set has the same range of ten and, later,

cardinal number of twenty. Place a set of objects before the child. when the

members are Note: The objects can be Ask: "How many in the set?"

re-arranged. similar or dissimilar.

** When the child has answered correctly, rearrange the objects, spacing them widely or closely or vertically or randomly or in any pattern.

Ask: "How many now?"

"Are there as many now as there were before?"

Re-arrange several times, on each occasion asking,

"How many now?"

"Are there as many now as there were before?"

Repeat, using other sets of objects, to give the required number of test items.

Acceptable Response:

The child must respond correctly to both questions without re-counting.

Note: The teacher may ask "How do you know?" The child should reply to indicate that no objects were added or taken away.

** The test proper begins here, OBJECTIVES. TEST MATERIALS TESTPROCEDURES and COMMENTS. The Child.

16. (a) Understands (a) For each test item two (a) i.PROCEDURE: and (b) uses the sets of objects with

teens "more" and different cardinal Place two sets before the child. "less" ("fewer") numbers. (through compari- Ask: "Which set has more members?" ("objects","thin son of sets). At first within the "Which set has fewer members?" range l10. "Which number is more?" Note: The word "less" "Which number is less?" used when referring Later, within the to quantities, the range 1.20. Repeat, using other set, two ata time, to give the required number of test items. word "fewer" whenre fining to numbers of Acceptable Responft: things (e.g. less butter, fewer apples). The child indicates the appropriateset, and states the correct number.

Note: The child may use counting, pairingor visual judgement in comparing the sets.

ii. For each test itema ii, PROCEDURE: set of objects, and a further collection of Place the set before the child. Give hima box containing further objects. objects from which the

child can make other Say: "Make a set withmore members than my set." sets. "Which number is more?" "Make a set with fewer members thanmy set," (The objectscan be "Which number is less?" white rods randomly

arranged or in line Repeat, beginning witha different set on each occasion, to give the required formation.) number of test items.

(Continued over) NUMBER

OBJECTIVES, TEST MATERIALS TEST PROCEDURES and COMMENTS,

The Child,.

16. (continued) It is suggested that in some repetitions white rods be used, randomly arranged, and in some, white rods in line formation, in the latter case telling the child to make a line with more rods instead of a set with more members,

Acceptable Response:

The child performs the appropriate action and indicates which number is more and which is less.

Note: The child may use counting, pairing or visual judgement to make his sets and to arrive at his answers.

iii. For each test item, iii, PROCEDURE: two rows of white rods of different Place two rows of white rods before the child, lengths (ends level) e.g. Ask: "Which row has more?" Then ask:"Which row has fewer?" (Two separate questions, to be answered separately.)

Repeat, using other pairs of prepared rows, to give the required number of test items.

Acceptable Response:

The child indicates the appropriate row, in response to each question. It should not be necessary to count the white rods (since the ends of the rows are level) but the child may do so if he needs to. NUMBER

TEST MATERIALS TEST PROCEDURES and COMMENTS.

1.1....

iv. For each test item, iv. PROCEDURE: two rods of different lengths (ends level). Place the two rods before the child, (Indicate white is the unit.) e.g. Ask: "Which is more?" Then ask:"Which is less?" (Two separate questions, to be answered separately.)

Repeat the test, using other rods, to give the required number of test items.

Acceptable Response:

The child indicates the appropriate rod, in response to each question.

(b) i,For each test item, (b) PROCEDURE: two sets of objects with different cardinal Place two prepared sets of objects before the child. numbers. (The sets can be of white rods Ask: "Which set has more and which set has fewer members? (One question.) randomly arranged, or in line formation, Repeat, to give the required number of test items, using other prepared sets two for some of tilt tests.) at a time. For some repetitions use white rods randomly placed, for others use ite rods in line formation, in the latter case asking which line has more rods, and which has fewer.

Acceptable Respon:

The child uses the words "more" and 'fewer" ("less") in his response, for example:

"The set has more; this set has fewer", indicating the appropriate sets,or

(Continued over) NUMBER

OBJECTIVES. TEST MATERIALS TEST PROCEDURES and COMMENTS. The Child

16, (continued) "This line has more rods, this line has fewer rods," indicating the appropriate lines,

If the child says:"This set has more; this set hasn't," he is at an intermediate stage.

Note: Accept the word "less" for "fewer" ina child's response,

ii. For each test item, ii. PROCEDURE: two rods of different

lengths (ends kvel). Place two different rods before the child. (Indicate white is the unit. e.g.

Ask: "Which is more and which is less?" (One question.)

Repeat, using other pairs of different rods,to give the required number of test items.

Acceptable Response:

The child uses the words "more" and "less" in hisresponse, for example:

"This one is more, this one is less," or "Seven is more, six is less,"

Note: A response of the type "This is seven, this is si:," (i.e. numbers only) is not acceptable. If the child responds in this way, repeat the question to elicit if possible a response containing the words "more" and "less". NUMBER

OBJECTIVES. TEST MATERIALS TEST PROCEDURES and COMMENTS.

The Child.

17. Understands that 1. Several equivalent sets, i. PROCEDURE: all sets whichare within the range 110. equivalent have Place before the child two of the prepared sets. the same cardinal (The sets need not be

number. arranged so as to show Ask: "How many members in each set?" equivalence.) "Which is more, which is less?"

The child indicates that both have the same number.

Place a third set beside the other two.

** Ask: "Is this more? Is it less?" "What can you tellme about the three sets?"

The child indicates that the three sets have the same number.

Repeat, to give the required number of test items, using therest of the prepared sets, two at a time.

Acceptable Response:

For each test item the child indicates that he realises the equivalenceby a response such as "They are all the same number." "They all have thelame number." "They are all sevens." If the child answers "No" to the questionegarding more, or less, say "Why do you think that?"

Note: Accept the child'sresponse if he uses "equal" or "the same number" to indicate equivalence,

" The test proper begins here,

(Continued over) NUMBER

TEST MATERIALS TEST PROCEDURES and COMMENTS.

. Several rows of white ii.PROCEDURE: rods all having the same number. Place before the child two of the rows of white rods, ends level,

Ask: "Which is the longer row?" "Which row has more?"

"Which is the shorter row?" "Which row has fewer?"

"The child replies "They are both the same", or "They both have the same number."

** Now place a third row beside the other two. OOOOO Ask: "Is this row longer?" "Shorter?" "Is this row more?""Less?"

The child indicates that the threerows have the same number.

Repeat to give the required number of test items, using the rest of the prepared rows of white rods,

Acceptable Response:

For each test item the child indicates that he realises the equivalence bya response such as "They are all the same number." "They all have the same number." "They are all sevens," If the child answers "No" to the questions regarding more, or less, say "Why do you think that?"

Note: Accept the child's response if he uses "equal" or "the same number" to indicate equivalence.

** The test proper begins here. Continued NUMBER

OBJECTIVES. TEST MATERIALS TEST PROCEDURES and COMMENTS.

The Child.

17,(continued) Hi. A number of rods of iii,PROCEDURE: the same colour.

Place before the childtwo of the rods, ends level.

Ask: "Which is the longer rod?" "Which is the shorter?"

"Which is more?" "Which is less?"

The child indicates the rodsare the same.

** Now place the third rod beside the othertwo.

Ask: "Is this rod longer?""Shorter?" Is this rod more?""Less?"

The child indicates the rod is thesame as the other two.

Repeat, to give the required number oftest items, using the rest of the selected rods.

Acceptable Response:

For each test item the child indicates that he realisesthe equivalence by a response such as "Theyare all the same number," "They all have the same number." "They are all sevens." If the child answers "No" to the questions regardingmore, or less, say 1.1,:y do you think that?"

Note: Accept the child's response if heuses "equal" or "the same number" to indicate equivalence.

* The test proper begins here. NUMBER

OBJECTIVES, TEST MATERIALS TEST PROCEDURES and COMMENTS. The Child

18. (a) Understands (a)Pairs of equivalent sets. (a) PROCEDURE: the meaning of, and (b) writes tt Place before the child two equivalent sets of seven members each. the equals sign. Ask: "How many in this set?" "And the other set?"

The child answers, "Seven" on both occasions.

Write:7 = 7

** Ask: "What does this number sentence say?" "What does it mean?" "What is this sign called?" (indicating the equals sign)

Repeat, to give the required number of test items, using different pairs of equivalent sets.

AcceptableResponse:

For each test item the chiki should be able to state that, for example

seven equals seven

the sets have the same number

the sign is called "equals".

" The test proper begins here. ttThe number may be varied, ri 4 I Continued La /. f. NUMBER

OBJECTIVES. TEST MATERIALS TEST PROCEDURES andCOMMENTS, The Child.

18. (continued) (b) i, Pairs of equivalent sets. (b) PROCEDURE:

Objects.

Place before the child two equivalentsets of eight members each.

Ask the child to count the objects and makethe appropriate recording.

Say: "Count these." "Write the numbersentence." or "Write it down theway I did."

Repeat, to give the required number oftest items, using other pairs of equivalent sets

Acceptable Response:

The child writes hisresponse in the form: 8 = 8

A box of rods. ii.Rods,

With lines of white rods.

tt Place before the child two linesof white rods with eight in each line.

Ask: "How many rods in this line?" "And this line?"

The child answers "Eight"on each occasion.

Write for the child:8 = 8

f The number may be vaned,

(Continued over

6 5 NUMBER

OBJECTIVES. TEST MATERIALS TEST PROCEDURES and COMMENTS.

The Child

18, (continued) Proceedas outlined in(a) and (b) above, making suitableadju'itmentsto the wording,

With pairs of rods of the same colour.

tt Place before the child two brown rods side by side. Ask the child to state the value of the rods (unit, white),

Then write:8 = 8

Proceedas outlined in(a) and (b) above, making suitableadjustments tc the wording.

Repeat, to give the required number of test items, using other numbers of white rods, and using pairs of rods of other colours.

Acceptable Responses:

(a) For each test item the child should be able to state that, for example,

eight equals eight

the rods, or lines of rods, represent the same number

the sign is called "equals",

(b)The child writes his response in the form

8 = 8

tt The colour may be varied. NUMBER

OBJECTIVES. TEST MATERIALS TEST PROCEDURES and COMMENTS. The Child

19. (a) Separates a set (a) i,Sets of objects with (a) i.PROCEDURE: into subsets, and members within the (b) understands range of ten, and Place one of the sets before the child. part-whole later, twenty. relationships. Say: "Break this set into subsets"or, "into separate subsets." "How many subsets didyou make?"

Repeat to give the required number of test items. Oneach occasion the child i: asked to repeat the action, breaking the set intoother subsets, and differetr numbers of subsets.

On each occasion ask: "Howmany subsets did you make?"

Acceptable Response:

The child uses all the members when making thesubsets, indicating that hi recognises thatevery member must belong to a subset. He states correctly the number of subsets made.

The child mayuse attributes such as colour, shape, size or position, when select: ing members for subsets;on the other hand he may now form subsets whom only attribute is that of cardinal number, the membersbeing quite dissimilar.

(a) ii. A box of rods. (a) ii.PROCEDURE:

Say: tt "Take outa yellow rocr (Specify yellow or longer.) "Make a pattern for this rod." "How many rods in the second line?"

"Now make a different pattern for it," "How many rods this time?"

f 't The colour may be varied,

(Continued over) NUMBER

OBJECTIVES. TEST MATERIALS TEST PROCEDURES and COMMENTS. The Child

19.(continued) Repeat, to give the required number of test items, On each occasion the child is asked to repeat the action, using in the second line different rods and different numbers of rods, and on each occasion being asked to state the number of rods in the second line.

Acceptable Response:

The child makes the patterns correctly and responds with the correct number.

(Although the word "subset" has not been used in this activity, the child has in effect illustrated and counted the subsets of the set, The original rod represents

the set, and the rods in the second line of the pattern represent thesubsets.)

Note: If it is desired to reinforce the concept of subsets in the context of rod illustrations then teaching, and testing, may include the following:

Give the child the original rod as above (i.e. yellow or longer),

Ask him to show the equivalent set of white rods, then to separate thest into subsets.

Request that a single rod be substituted for each subset of white rods.

Form the rod pattern as in (a) ii above or invite the child to doso.

Ask: "How many subsets did you make?"

Continued OBJECTIVES, TEST MATERIALS TEST PROCEDURES arid COMMENTS, The Child

?, (continued) (b) A collection of coloured (b) PROCEDURE: beads.

i.Place before the childa set of ten beads six red, one green, one pink, one blue, A collection of coloured one yellow. counters.

Say: "Look at these beads." A collection of coloured

buttons. Ask: "How may beadsare there altogether?" "How many red beads are there?" (Other objects may be

used, the wording in ** "Are there more beads than red beads ?" the tests being varied

accordingly.) ii.Place before the child a set of ten beadssix red, four blue.

Ask: "How many beadsare there altogether?" "How many red beadsare there?"

** "Are there more beads that red beads?" Now: These tests use the

attribute of colour. If desired, iii.Place before the child a set of teb beadssix red, four blue. other attributes e.g. shape,

size, texture, may be used, Ask: "How many beadsare there altogether?" and the wording varied as "How many red beadsare there?" required.

** "Are there more red beads than beads?"

** The test proper begins here, (Continued over)

6 r) NUMBER

OBJECTIVES. TEST MATERIALS TEST PROCEDURES and CO-Vir The Child. i9, (continued) iv.Place before the childa set of ten countersfour 'red, four green, two blue,

Ask: "How many countersare there altogether?" "How many green countersare there?"

** "Are there more green counters than counters?"

v.Place before the childa set of ten buttons three green, three yellow, three red, one blue.

Ask: "How many buttons are there altogether?" "How many yellow buttonsare there?"

** "Are there more yellow buttons than buttons?" "Are there more buttons than yellow buttons?"

If ten items are desired, five futhertest items of a similar nature may be devised,

Note: With older children the first two questions should beomitted on each occasion, and the third question only should be asked.

With younger children the first twoquestions may be asked, to ensure that the child is not confused by the wording of the final question. Itis found that often children tend to answer the question they think is being asked,for example, in ii, "Are there more blue beads than red beads?"

If a child succeeds in a number oftest items using the three questions, he should then be given a number of othertest items of a similar nature using questions of the third type only.

** The test proper begins here. Continued NUMBER

OBJECTIVES. TEST MATERIALS TEST PROCEDURES and COMMENTS.

The Child. ,

19. (continued) Acceptable Response:

The child answers correctly the third question (marked **) in each test item to indicate that he understands that the whole is greater than the part, or that the cardinal number of the set is greater than that of the subset. (The part is less than the whole, the cardinal number of th! subset is less than that of the set.)

He may state merely "yes" or "no" in answer to the question, or may answer in the form

"Yes, because there are ten beads, and four red beads."

If desired, when a child has answered just "yes", or "no", the fester may continue:

"How do you know?" or "Why do you think that?"

The child may then indicate that he understands the relationship of the whole to the part by statements such ris the following:

"There are ten beads, but only four blue beads." "First youmean all the counters,then you look at justt? :e red ones." NUMBER

OBJECTIVES. TEST MATERIALS TEST PROCEDURES and COMMENTS. The Child

20. Knows that zero A set of three objects and i, PROCEDURE: is the cardinal a set of numeral cards number and 010. Place the set of objectsbefore the child, and give him the numeral nought is the cards. numeral for the Say: "Howmany in this set?"(three) set that has no "Find the right numeral card,"(3) members. "What is its name?" (three)

Remove'one object, "How many in this set?" (two) "Find the right numeral card."(2) "What is its name?" (two)

Remove one object, "How many in this set?" (one) "Find the right numeral card,"(1) "What is its name?"(one) Nate: The cardinal ** Remove the last object. numbers and their "How many in this set?"(zero, none, nothing) associated numerals "Find the right numeral card."(0) have the samenames "What is its name?"(nought) except in the case of the empty set, i.e. If a child responds bysaying "none", "nothing", "empty the cardinal number set", ask "Is there another word?" is zero and the name of the numeral is Acceptable Response: nought

The child respondsas outlined, (This is the convention defined in the N,S,W, Note: It is not possibleto repeat this procedure to give five mathematics syllabus.) or more test items in different forms. The onlyvariation possible is to use differentsets of three objects.

**The test proper begins here, Continued NUMBER

OBJECTIVES. TEST MATERIALS TEST PROCEDURES and COMMENTS. The Child. *.r...

20. (continued) ii. A box of rods. itPROCEDURE:

It is not vital that Pencil and paper Place the incomplete pattern before the child. Give him pencil andpaper and have children maintain the available a box of rods, State that white is the unit, distinction between An incomplete pattern the names nought and for the orange rod: Say: "Complete the pattern."(red rod) zero, merely that they orange side by side "What number goes with eight to make ten?"(two) are conversant with with brown "Write it down."(2) them. "What is it called?"(two) The concept of the empty set, however, Replace the brown rod with blue, is important. "Complete the pattern."(white rod) "What goes with nine to make ten?"(one) "Write it down."(1) "What is it called?"(one)

**Replace the blue rod with an orange rod,

Say: "What goes with ten to make ten?"(nothing)

When the child says "nothing" or "no rod", ask "Is there another name?"(zero).

Continue: "Write it down,"(0) "What is it called?"(nought)

Acceptable Response:

The child responds as outlined.

Note: It is not possible to repeat this procedure to give five or more test items in different forms. If varied by selecting other rods as the starting rod, the test is merely lengthened.

$$ The test proper begins here, NUMBER

OBJECTIVES. TEST MATERIALS TEST PROCEDURES and COMMENTS. The Child

21, (a) Combines two (a) i.Several sets of objects (a) i.PROCEDURE: sets to form a (with appropriate new set and numbers of members). Place two of thesets before the child, (b) understands that this is an Say: "Combine thesetwo sets."("Put together, action of "How many in thenew set?" addition.

Repeat, to give the required number oftest items, using other pairs of sets.

Acceptable Response:

The child performs the action, andstates the nun correctly.

ii. A box of rods. ii. PROCEDURE:

Place the box of rods before the child.Give the child two suitable rods.

Say: "Place these rods endtoend,"

"Make a pattern." (that is,measure the rods). "Which rod did youuse to make the pattern?"

Repeat, to give the required number oftest items, using other pairs of rods,

Acceptable Response:

The child makes thepattern correctly, andnames the rod which equals the length of the nominated rod.

Continued NUMBER

TEST MATERIALS TEST PROCEDURES and COMMENTS.

(b). 1.Several sets of objects (b) i.PROCEDURE: (with appropriate numbers of members) Place two of the sets before the child. as in (a) i above. Say: "Combine these two sets," "Tell me the number sentence,"

Repeat, to give the required number of test items, using other appropriate sets.

Acceptable Response:

The child gives the correct number sentence.

(b) ii. A box of rods. (b) ii. PROCEDURE:

Place the box of rods before the child. Give him two suitable rods.

Say: "Place the rods endto-ind," "Make a pattern and tell me the number sentence,"

Repeat, to give the required number of test items, using other appropriate rods.

Acceptable Response:

The child states the number sentence in a form similar to one of the following:

"Three plus two equals five." "Three and two equals five." "Two added to three is equal to five."

Note: It is assumed that the terms "adding" and "addition" have been used ini the teaching process.

(Continued over) NUMBER

OBJECTIVES. TEST MATERIALS TEST PROCEDURES and COMMENTS. The Child

21.(continued) The teacher may continued thetest procedure by asking:

Note:Regardless of "What have we been doing?" or the number of addends "What action is that?" or the actual process of "What does plus tell us?" addition involves combining only two The child should be able to respondby saying, "Adding" or "Addition". sets at a time.

Tests may be given wherea further set is combined with the "new set" previously made.

el, (3 + 2) + 4 9

i. With sets of objects.

Give the childa set of three and a set of two; when he has combined and counted these,give him a further set of four, Then ask for the total and thenumber sentence.

ii. With rods,

Give the child two rodsto measure; ask for the total and the number sentence. Then give the thirdrod, to be placed end -to- end with the first two. Ask forthe new number sentence,. NUMBER

TEST MATERIALS TEST PROCEDURES and COMMENTS. Thy vtuId

22. Understands that Four sets of objects i. PROCEDURE: whatever the the total notto exceed order in which ten. Place the four sets before the child, sets are combined

the total number Ask: "Howmany in these sets altogether?" of members is

the same. The child responds by giving the correct number,

* Place thesets in a ient ord 'nd say:

"Now how many?"

Continue tore-arrange the order of the sets, and on each occasion ask:

"Now how many?"

Repeat, to give the required number of test items, using four differentsets on eacl occasion.

kceptable Response:

The child gives the correct number without recountingthe members.

ii.A, box of rods, ii. PROCEDURE:

Place before the child four rods end-to-end (totallength not longer than orange} Indicate the unit in the number stage.

Say: "Measure these rods."(with one rod) "How long are they?"or "What do they equal?"

** The test proper begins here. (Continued over: NUMBER

OBJECTIVES. TEST MATERIALS TEST PROCEDURES and COMMENTS. The Child.

22. (continued) The child responds in colouror in number, according to stage, For example, "Blue" in prenumber stage, or "Nine" in number stage.

** Remove the measuring rod,rearrange the original four rods, then ask:

.`Not long?"or "Now how many?"

Continue to change the order of the rods, asking thesame question after each rearrangement.

Repeat, to give the required number of test items, using fourdifferent rods on each occasion.

Acceptable Response:

The child responds correctly without remeasuringthe rods.

23. (a) Separatesa (a) i. Sets of objects. (a) i.PROCEDURE: set into two sub-

sets, removes one Place a set of objects before the child. subset, and (b) understands Ask: "How manyare there?" that this is an "Separate this set intotwo subsets and remove one subset." action of subtraction, Ask: "Howmany did you remove?" "How manyare left?"

Repeat, to give the required number oftest items, using other sets of objects.

** The test proper begins here. Continued NUMBER

TEST MATERIALS TEST PROCEDURES and COMMENTS.

Acceptable Response:

The child carries out the actions and states the numbers correctly.

(a) u, A box of rods. (a) ii.PROCEDURE:

Place one rod before the child (preferably a rod longer than yellow). Indicate the unit.

Ask: "What rod is this?""'hat is its number?" Say: "Make a pattern for it using two more rods."

The child selects two rods and makes a pattern.

Say: "Place one of the shorter rods on top of the long one." Ask: "What is the number of the rod on top?" "What is the number of the rod which fits in the space?"

Repeat, to give the required number of test items, using oth' -id patterns.

Acceptable Response:

The child carries out the actions and states the numbers correctly.

(b) i.A set of objects. (b) i.PROCEDURE:

Place the objects before the child,

Ask: "How many are there?" Say: "Separate this set into two subsets and remove one subset,"

Say: "Tell me the number sentence." (Continued over) NUMBER

OBJECTIVES. TEST MATERIALS TEST PROCEDURES and COMMENTS. The Child...

23. (continued) Repeat, to give the required numberof test items, using othersets of objects.

Acceptable Response:

The child states the numbersentence in a form similar toone of the following:

"Seven minus two equals five," "Seven take away two is five."

(b) ii. A box ofrods. (b) ii. PROCEDURE:

Place one rod before the child (preferably a rod longer than yellow). Indicatethe unit, (white).

Ask: "What rod is this?""What is its number?" "Make a pattern for it using twomore rods?'

When the child has made thepattern, say:

"Place one of the shater rodson top of the long one." "Tell lie the number sentence."

Repeat, to give the required numberof test items, using other rods andpatterns.

Note: If the child selects the secondrod to place on top, the conventional subtraction structure willappear in reversed form.

Acceptable Response:

The child states the numbersentence in a form similar toone of the following:

Continued 1.J NUMBER

OBJECTIVES. TEST MATERIALS TEST PROCEDURES and COMMENTS. The Child.

23. (continued) "Seven minus two is five." "Seven minus two equals five."

Note: Itis assumed that the terms "taking away" "subtracting"and "sub traction" have been used in the teachingprocess. The teacher may continue the test procedure by asking:

"What havewe been doing?" "What action is that?" or "What does minus tell us?"

The child should be able to respond by saying "Takingaway", "Subtracting", or "Subtraction",

24. Compares two i. Sets of objects with i. PROCEDURE: sets bypairing to different numbers of find the difference members. Place two sets of objects before the child, eachset containing a different number of between them, (not more than ten objects. and understands per set) that gils is also Say: "Sec if these sets match." (or "Pair the elementsof these sets.") an action of Ask: "Are they thesame in number or are they different?" subtraction, "What is the difference ?""Tell me the number sentence."

Repeat, to give the required number of test items,using other sets presented two at a time.

(Continuedover) NUMBER

TEST MATERIALS TEST PROCEDURES and COMMENTS,

11111

Acceptable Response:

The child carries out the action of pairing the members of thesets and noting tht objects unpaired, he states the correct numbers, and he gives thenumber sentence correctly in subtraction form, If hecan state the difference but cannot give the subtraction sentence he is at an intermediate stage.

Note: It is assumed that the terms "difference", "comparing","subtracting" and "subtraction" have been used in the teachingprocess, The teacher may continue the test procedure by asking:

"What have we been doing?" "What action is that?" or "What does minus tell us?"

The child should be able to respond by saying "Comparing", "Subtracting""Sub- traction" or "Finding the difference",

ii.Pairs of rods of different u.PROCEDURE: lengths.

Place before the child two unequal rods,

Say: "Put these two rods side by side." Ask: "Are they the same or are they different?"

Indicate that white is the unit,

Ask: "What is the difference?""Tell me the number sentence."

Repeal, to give the required number of test items, using other pairs ofunequalrods,- 3::, Continued NUMBER

OBJECTIVES. TEST MATERIALS TEST PROCEDURES and COMMENTS. The Child

(continued) Acceptable Response:

The child carries out the action of comparing rods, andnoting or measuring the length of the "gap", he states thecorrect number for the difference, and he gives the correct number sentence in subtraction form.

Note: The test may be extendedas explained in the Note for i. above,

,Understands that i. An assortment of objects. i. PROCEDURE: there is an inverse relationship Place a set of seven objects before the child, (Donot actually add or remove any between the objects during the course of the testing,as the test is of a mental process or under- operations of standing.) addition and

subtraction, and Ask: "How many in the set?" of subtraction "How many would there be if I added three and thenremoved (subtracted) and addition, the same three?"

"How many would there be if I added three and thenremoved (subtracted)! three others?"

"How many would there be if I removed (subtracted) threeand then added the same three?"

"How many would there be if I removed (subtracted) three andthen added three others?"

Repeat, to give the required number of test items, using othersets and numbers,

(Continued over) NUMBER

OBJECTIVES. TEST MATERIALS TEST PROCEDURES and COMMENTS.

The Child.

25. (continued) Acceptable Response:

The child responds by answering "seven" to all the questions (in theexample cited) and other appropriate numbers (to all the questions in eachtest item) without carrying out the physical actions with objects.

The teacher may continue each test item by asking: "How doyou know?" The child may respond with answers suchas:

"It's the same as we started with." "If we add something and then take itaway it makes no difference." "If we add a number and subtract thesame number it makes no difference." "If you take something away and then put it back, it's thesame as it wasat first."

Any response is acceptable which shows that the child understands thatone operation "undoes" the other.

ii.A box of rods. PROCEDURE:

Give the child a black rod. Indicate that white is the unit.

(Do not actually add or removeany rods during the course of the testing, as the rest is of a mental process or understanding.)

Ask: "What number does this rod represent?"

ft "What would be the answer ifwe add a lightgreen rod and subtract a lightgreen rod?"

"What would be the answer if we subtracta lightgreen rod and a lightgreen rod?"

ttThe colour may be varied, Continued NUMBER

OBJECTIVES, TEST MATERIALS TEST PROCEDURES and COMMENTS, The Child

25, (continued) Repeat, to give the required number oftest items, using other rods and numbers.

Acceptable Response:

The child responds "seven" (in theexample cited) without carryingout the physical actions with rods, The teachermay continue each test item by asking:

"How doyou know?" The child may respond with answers suchas:

"It's thesame rod we started with," "If we adda rod and subtract the same rod it makes no difference."

Any response is acceptable which showsthat the child understands thatone opera- ticn "undoes" the other.

26, Understands that i. Sets of objects. i. PROCEDURE: from one addition (up to twenty objects, fact, other preferably identical) Place before the childa set of seven objects. addition facts and subtraction facts Say: "Separate thisset into two subsets." can be derived. "Tell me the number sentence,"

"Tell me as many other number sentences about thisas you can think of." (Itmay be necessary to prompt to elicit silbtraction sentences.)

Repeat, to give the required number oftest items, starting on each occasion witha set containing a different number of members.

** The test proper begins here, (Continuedover NUMBER

TEST MATERIALS TEST PROCEDURES and COMMENTS.

Acceptable Response:

After giving his initial number sentence correctly, for example "seven equals five

plus two", the child gives other number sentences derived from it, suchas:

7=2+5 7 2 5 5+2=7 7 5 = 2

2+5=7 2 = 7 5 7-5-2=0 5 = 7 2

The minimum requirement forsuccess in each test item is two addition and two subtraction forms,

Note: For the derived number sentences, the childmay manipulate the objects, or may work mentally. He may state the number sentences orally, or give them in written form. He may write his first number sentence, and state the others orally as he refers to theirecording.

A box of rods. ii. PROCEDURE:

Place before the child a brown rod.

Say: "Make a pattern for the brown rod, using two rods in the second line." "Read the pattern."(The reading will be given in colour for prenumber stage, in number for number stage.) ** "Tell me as many other sentences about this patternas you can think of."

Repeat, to give the required number of test items, using other rods andpatterns,

s" The test proper begins here, Continued NUMBER

OBJECTIVES, TEST MATERIALS TEST PROCEDURES and COMMENTS. The Child

26. (continued) Acceptable Response:

The child responds by giving, in the colourstage, sentences such as the following, all derived from an initial sentence, "brown equals yellow pluslight-green":

"Yellow plus light-green equals brown." "Light-green plus yellow equals brown." "Brown equals light-green plus yellow." "Brown minus lightgreen equals yellow." "Brown minus yellow equals light-green." "Yellow equals brown minus lightgreen," "Light-green equals brown minus yellow." "Brown minus lightgreen minus yellow equals nothing."

and sentences, in the number stage, such as:

5 + 3 = 8 3+5=8 8=3+5 8-3=5 8-5=3 5=8-3 3=8-5 8-3-5=0

The minimum requirement forsuccess in each test item is two addition and two subtraction forms.

Note: The child may manipulate the rods, or make mental transpositions. He may state the number sentences orally, or give them in written form. He may write his first number sentence and state others orallyas he refers to his recording, NUMBER

OBJECTIVES, TEST MATERIALS TEST PROCEDURES and COMMENTS. The Child.. ,

(a) Combines (a) i.A large collection of (a) i.PROCEDURE: several sets, each objects from which to with the same make equivalent sets. tt Place before the child foursets, each with two members, cardinal number, to form a new Say: "Combine these four sets to makea new set." set, and "How many in thenew set?" (b) understands that this action Repeat, to give the required number oftest items, using other groups of of repeated equivalent sets, addition is multiplication, Acceptable Response:

The child performs the actions and states thenumbers correctly.

(a) ii.A box of rods. (a) ii.PROCEDURE:

tt Place before the child four red rods,

Say: "Place these endto-end and then complete thepattern." "What do they measure?"

Repeat, using other sets of rods of thesame colour, to give the required number of test items.

Acceptable Response:

The child makes the pattern correctly andgives the correct value of the measuring rod,

tt The number may be varied, b Continued OBJECTIVES. TEST MATERIALS TEST PROCEDURES and COMMENTS. The Child

..101.1nmkimMela

27. (continued) (b) i,A large collection of (b) i.PROCEDURE: objects from which to

make equivalent sets. Carry out the procedure (a) i.

Say: "Tell me the number sentence."

Repeat, using other sets and numbers,to give the required number of test items.

Acceptable Response;

The child states the numbersentence in a form similar to one of the following:

"Four twos are eight." "Four twos equal eight." "Four lots of two equal eight." "Four times two equals eight."

Note;If the childuses an addition form say, "Tell me another way."

(b) 1 A box of rods. (b) 1 PROCEDURE:

Carry out procedure for (a)

Say: "Tell me the number sentence."

Repeat, using other "train"patterns, to give the required number of test items.

AcceptableResponse:

The child states his numbersentence in a form as indicated in (b) i.

S NUMBER

OBJECTIVES. TEST MATERIALS TEST PROCEDURES and COMMENTS. The Child

28. (a)Partitions a (a) i.A collection of objects. (a) i.PROCEDURE: set into two

equivalent sub Place before the child a set of objects withan even number of members, sets, and

(b) understands Say: "Separate this set into two subsets each with thesame number."(or,. that this is the "into two equivalent subsets.") action for

finding halves. Ask: "How many in each (subset)?"

Repeat, to give the required number of test items, using other sets which have even number of members. Note: Procedure (b)i may be presented Acceptable Response: immediately after

(a) i if desired, and The child performs the action and states the number correctly. Procedure (b) ii may be presented immediately after (a) E. (a) 1 A box of rods. (a) I PROCEDURE:

Place before the child a rod representingan even number. (Indicate white :s th unit.)

Say: "Make a train of two rods for this rod." Ask: "What colour is the train?" "What number is each train rod?"

Repeat, using other rods representingeven numbers, to give the required numbe of test items.

Continued NUMBER =mdm. .=ra TEST MATERIALS TEST PROCEDURES and COMMENTS,

Acceptable Response:

The child makes the train and states the number correctly.

(b) i.A collection of objects. (b) i.PROCEDURE:

Carry out the procedure for (a) i.

Ask: "What didwe do to the set?"

"What have we found out about the set?" or "Tell me something about the subsets." or "Tell me something about the parts of the set."

Then say:

"Tell me a number sentence about whatwe did."

Repeat, using other sets which havean even number of members, to give the required number of test items.

Acceptable Response:

The childmay respond in any way similar to the following:

"We halved the set." "We found the two halves of the set." "Each part is a half." "Five is half of ten."(for example).

If the childsays: "We made two subsets with thesame number" (or "two equivalent subsets"), continue questioning to ascertain whether he knows the terms half, halves or halving.

(Continued over) NUMBER

TEST MATERIALS TEST PROCEDURES and COMMENTS.

(b) IL A box of rods. (b) ILPROCEDURE:

Carry out the procedure for (a) ii.

Ask: "What did we do to the roaror "What have we found out about it?"or "Can you tellme something about each train rod?"

Then ask: "Can you tell mea number sentence about this?"

Repeat, using other rods which have two-rod trains, to give the required number of test items,

Acceptable Response:

The child may respond inany way similar to the following:

"We halved the rod."

"We found thetwo halves." "Each of these rods isa half," "Five is half of ten," (for example)

If the child makes a response such as:"We made a yellow train", continue

questioning to ascertain whether he knows the terms half, halvesor halving, NUMBER

OBJECTIVES. TEST MATERIALS TEST PROCEDURES and COMMENTS. The Child

29. (a) Understands (a) i,Objects from which (a) i, PROCEDURE: and (b) uses the sets can be formed,

terms odd and Place the objects before the child. even,

Say: "Make me a set with an odd number of members." "Now make me another 'odd' set,"

Repeat the instruction, to give the required number of test items,

Acceptable Response:

Note:If desired, the The child makes sets with odd numbers of members. test items for the term even may be Note: The teacher may continue by asking: "How doyou know that is an presented before odd number?" The child indicates his understanding bya response similar to those for odd, the following:

"I can't put the set in twos, there's alwaysa two not finished." "There is one over if I put the set in pairs." "You can't get half of that set" (meaning, ofcourse, that there is no whole number which is half ofan odd number),

(a) ii,A box of rods. (a) ii,PROCEDURE:

Place the box of rods before the child. Indicate that white is theunit,

Say: "Show me a rod for an odd number," "Now show me another (foran odd number)."

(Continued over) NUMBER

^v.

TEST MATERIALS TEST PROCEDURES and COMMENTS.

.1,m1010111.=1

Repeat to Ole the required number of test items, After each rld is shown,say: "What number does that rod represent?" "How do you know that number is odd?"

Note: If ten test items are required, the child will have to show some of the rods more than once.

Acceptable Response:

The child selectsa white, light-green, yellow, black or blue rod and states the numerical value. He indicates his understanding by a response similar to the following:

"You can't make a red train for that rod." "You can't find two rods the same colour to equal that length."

(a) iii. Objects from which (a) PROCEDURE: sets can be made. Place the objects before the child.

Say: "Makeme a set with an even number of members." "Now make me another 'even' set,"

Repeat to give the required number of test items.

Acceptable Response:

The child makes sets with even numbers of members.

Coniinued NUMBER

TEST MATERIALS TESTPROCEDURES and COMMENTS. rmgy.,1=,

Note: The teachermay continue by asking: "How do you know that is an. even number?" The child indicates his understanding bya response similar to the following:

"I can put the set in pairs (twos)." "There isnone over when I make pairs." "I can find half of, that set." (indicating that he knowsthat half an even. number is alwaysa whole number),

(a) iv, A box of rods. (a) iv. PROCEDURE:

Place the box of rods before the child. Indicate thatwhite is the unit.

Say: "Show mc a rod foran even number." "Now hewme another (for an even number)."

Repeat to givf: the required number of test items. Aftereach rod is shown say:

"What number does that rod represent?" "How doyou know that number is even?"

Note: If ten test itemsare required, the child will have to show some of the rods more than once.

Acceptable Response:

The child selects a red, crimson, darkreen, brownor orange rod and states the numerical value, He indicates his understanding byresponses similar to tilt following:

"You can makea red train for that rod." "You can find two rods thesame colour to equal that length."

(Continued over) NUMBER

OBJECTIVES, TEST MATERIALS TEST PROCEDURES and COMMENTS. The Child., .

29. (continued) (b) i.Several sets (five or i. PROCEDURE: ten) some containing

odd numbers of Place the sets before the child, members, some

containing even Say: "Tell me which setsare odd 'and which sets are even." (One instruction, numbers of members. not two.)

Acceptable Response:

The child points to each set in turn anduses the wordsoddandeven, saying, for example:

"This set (number) is odd and thisset (number) is even."

Note: The child may arrange the objectsor count them.

(b) ii,Several rods (five or (b) ii,PROCEDURE: ten), some represent

ing odd numbers, Place the rods before the child, Indicate that white is the unit, some representing even numbers, (Some Say: "Tell me which rods stand for (represent)even numbers and which stand colours may be for odd numbers." (One instruction, not two.) include twice.)

Acceptable Response:

The child points to each rod inturn, and uses the wordsoddandevensaying, for example:

"This number is odd and this number is even."

Continued NUMBER

TEST PROCEDURES and COMMENTS.

29. (continued)

Extension: Extension:

Recognises odd and A pencil i.PROCEDURE: even numbers.

A variety of cards Place a card before the child. If a child shows with numeralsas shown:

mastery of the Say: "Circle all the numerals for odd numbers." (or "all the odd numbers.") work outlined in In the range 1-10: (a) and (b) above, e.g. Show further cards for the odd numbers, to give therequired number of test items, this extension first in the range 1.10 and, later, 1.20. may be Oven. 4 5103 8 5 3 1 7 Acceptable Response: 9 57636 6a 42 149 The child circles all the numerals for odd numberson the card, for each test item. 4 3 8 2 3 10

In the range 1-20: 1 PROCEDURE; e.g.

Repeat the procedures outlined for the odd numberssubstituting the word "even" 4 18 2 10 15 6 for "odd" throughout.

11 17 16 20 14 Acceptable Response: 19 720 17 18 13

1312 5 12 19 10 The child circles all the numerals foreven numbers on the card, for each test item. NUMBER

OBJECTIVES. TEST MATERIALS TEST PROCEDURES and COMMENTS. The Child

A. (a) Partitions a A collection of objects. PROCEDURES: set into three,

four, five, , i, Thirds, equivalent sub.

sets and Repeat the procedures outlined in 28 (a)i, (b)i, (a)li, (b)ii, adapting the wordstc (b) understands give directions for sets with numbers divisible by three whichare to be partitione this as the action into three equivalent subsets, each of these being a third of the set,so that the child for finding may make a statement in a form similar to one of the following: thirds, quarters,

fifths, . "We separated the set into thirds," "We found the three thirds of the set," "Each part is a third," "Three is a third of nine," (for example).

Quarters,

Repeat the procedures outlined in 28 (a)i, (b)i, (b)ii, adapting the words tc give directions for sets with numbers divisible by four to be partitioned into foul subsets, each of which is a quarter (a fourth) so that the child may state:

"We separated the set into quarters," "We found the four quarters of the set." "Each part is a quarter," "Two is a quarter of eight."(for example).

Fifths,

Repeat the procedures outlined in 28 (a)i, (b)i, (b)ii, adapting the words tc

give directions for sets with numbers divisible by five whichare to br;artitioned into five subsets, each of these being a fifth, so that the child may state:

Continued NUMBER

TEST MATERIALS TEST PROCEDURES and COMMENTS.

"We separated the set into fifths." "We found the five fifths of the set." "Each part is a fifth." "One is a fifth of five,"(for example).

Sixths, sevenths, eighths, ninths, tenths.

As for fifths above, substituting six and sixths, seven and sevenths, eight and eighths, nine and ninths, ten and tenthsas required, where the words five and fifths are given.

Acceptable Response:

The child responds as indicated above,

Note: Children frequently find difficulty in separating a set into equivaleni subsets, and may instead make the required number of subsets with differeni numbers of members. For example, 4, 3 and 2 instead of 3, 3 and 3,

This happens when the child has no remembered number fact to use and "trial and error" is unsuccessful.

It is desirable to teach children a reliable technique for forming equivalent subsets.

This is the "one for you, one for you, one for you,., ," sharing technique,

Use the appropriate number of boxes or other containers, The child should place one object in each box, then repeat the procedure, one object to each box, until all the objects are distributed. He should understand that for success each "round" must be completed and there must be no "remainder" objects. NUMBER

OBJECTIVES. TEST MATERIALS TEST PROCEDURES and COMMENTS. The Child

31. (a) Partitions a (a) i,A collection of Six shares set into a given objects, number of sub, (a) PROCEDURE: sets to find out how many in Place before the childa set of objects whose number is divisible by six. each subset, and (b) understands Say: "Separate this set into six equivalent subsets" (or,"six subsets with the this as the action same number in each.") of sharing.

Ask: "Howmany in each subset?"

Repeat using other appropriate sets, and othernumbers of subsets ( "even" division) to give the required number oftest items.

(See Note,page 87) This is the form of

division most Acceptable Response: commonly used in problesolving The child makes correctly the specified number ofsubsets, and states the num situations, in the ber of objects in each. early stages.

Note: The child may use the technique explained in the Noteon page 85.

(a) ii. A box of rods. (a) ii.PROCEDURE: Note: Procedure (b) i may be present. Place before the childa rod, or rods, representing a number divisible by six. ed inunediately (Indicate that white is the unit.) after (a) i if desired Procedure (b)iimay Say: "Makea train with six rods in it for this rod" (or "for these rods"). be presented inuned. "What is the number of each trainrod"or lately after (a)ii if "Whatcolouriseachtrain.rod?" "Whatnumberisthat?" desired, NUMBER

TEST MATERIALS TESTPROCEDURES and COMMENTS.

Repeat, using other appropriate rods for which trains witha specified number of rods can be made, to give the required number of test items.

Acceptable Response:

The child makes the required train and states the number value of each rod in it.

Note: The procedures used in Objective 31 involve the same activities as used in Objective 30. In Objective 30, the understanding is of fractional parts, Here in Objective 31, the understanding is of sharing.

(h) i.A collection of (b) 1.PROCEDURE: objects. When the child has carried out the procedure with objects as outlined in 31 (a)i above, say:

"If these objects were lollies, and we shared them amongst six children how many would each child get?"or "What would each child's shar be?" or "Show me whatone person's share would be." "Can you tell me a number sentence for this sharing story?"

Acceptable Response:

The child observes the six subsets and states the number in each, He states th number sentence in a form similar to the following:

"Six twos equal twelve."or "Twelve has six twos,"

(Continued over NUMBER

OBJECTIVES. TEST MATERIALS TEST PROCEDURES and COMMENTS. The Child

Ell..114..1011111M 31. (continued) (b) ii. A box of rods. (b) ii.PROCEDURE:

When the child has carried out the procedure with rods, outlined in 31(a): point to the rod or rods for which the train hasbeen made and say:

"If this rod (these rods)were lollies and we shared them among six chile ren, how many would each child get?"or "What would each child's share be?"or "Can you tell me a number sentence forthis 'sharing' story?"

Acceptable Response:

The child observes the six rods in the train and states the number value ofeach He gives the number sentence in the form "Six twos equal (are)twelve" a "Twelve is six twos."

Five shares,

PROCEDURE:

Repeat all the activities outlined in (all, (a)ii,(b)i, (b)ii, using sets of object whose number is divisible by five,or rods representing numbers divisible by five The child mayuse the sharing technique outlined in the Noteon page 85.

Acceptable Response:

(a) The child makes the five subsets and states the number in each. (b) Hegive: the number sentence ina form similar to the following:

"Five twos equal ten."or "Ten has five twos." NUMBER

OBJECTIVES. TEST MATERIALS TEST PROCEDURES and COMMENTS. The Child ...

31. (continued) Three shares, two shares, four shares,...

PROCEDURE:

Adapt all the activities so that the numbers used are divisible by three, cr two, four or ... The child may use the sharing technique suggested on page 85.

Acceptable Respocae:

As outlined for other numbers of shares, substituting appropriate numbers.

Extensions: (c) PROCEDURE:

(c) Verbalises (c) The structures made by Follow up the procedures outlined by asking the child to explain his answer. this understanding. the child in (a) and (b) above. With sets of objects In response to the question "How do you know what each person's share is?" tl child should indicate that he understands that each subset corresponds to a persc and the number in the subset represents the person's share.

With rods.

In response to the question "How do you know what each person's share is?" t child should indicate that he understands that each train-rod corresponds to person, and that the number value of the train-rod represents the person's share.

(Continued ove NUMBER ,.....11.111N1111111011.01 OBJECTIVES. TEST MATERIALS TEST PROCEDURESandCOMMENTS. The Child.

31,(continued) Acceptable Response:

The child respondsas indicated.

Note: Although the procedures forvarious numbers of sharesare set out, it is no- necessary to give a set of test items for each number ofshares (i.e. test itemsor six shares, test items on four shares andso on). The set of test items given shoulc involve a variety of numbers.

32. (a) Partitions a (a) i. A collection of (a) i.PROCEDURE: set into equiva objects. lent subsets of Place before the childa set of object, whose number is divisible by three. a given number, to find out how Say: "Separate this set into subsets of three." many subsets are "How many subsets didyou make?" or "groups of three." formed and (b) understands Repeat, using sets with numbers divisibleby four, two, ,to give the required this as an action number of test items, of division,

Acceptable Response:

The child partitions the set correctly and states the number of subsetsmade.

lo Continued NUMBER

OBJECTIVES. TEST MATERIALS TEST PROCEDURES and COMMENTS, The Child...

32, (continued) (a) ii,A box of rods. (a)1 PROCEDURE:

This is the form of Place before the child a rod, or rods, representing a number divisible by three, division which asks (Indicate that white is the unit.) (for example) "How many threes ttSay: "Make a lightreen train for that rod (those rods)." in twelve?" (Four "How many rods in the train?" threes in twelve)

Repeat, using other rods representing numbers divisible by four, two, ,to give the required number of test items. Note: Procedure (b)i may be Acceptable Response: presented after (a)i if desired The child makes the train correctly and states the number of rods in the train. Procedure (b)ii may be presented (b) 1.A collection of (b) i.PROCEDURE: after (a)ii if objects. desired. When the child has carried out the procedure outlined in 32 (a)i say:

"How many threes in that number?"

Repeat this form of questioning throughout the required number of test items.

Acceptable Response:

te child observes the subsets and states, for example,

"There are four threes in twelve."

tt The colour may be varied. (Continued over) NUMBER 4wwwrorm

OBJECTIVES. TEST MATERIALS TEST PROCEDURES and COMMENTS. The Child

32. (continued) A box of pis. (b) ii. PROCEDURE:

When the child has carried out the proceduresoutlined in 32 (a)ii ask:

"Howmany threes in that number?"

Repeat this form of questioning throughout therequired number of test items. Note: Thisis the Acceptable Response: form of division commonly used in The child observesthe rods in the train and testing multiplic.ition states, for example: facts andin "There algorisms. are four threes in twelve."

33. (a) Removes (a) i. A collection of PROCEDURE: from a set equiv objects. alent subsets Place before thechild a set of objects whose wit.: a given number is divisible by four. number ofmem- Say: "Remove (asubset of) four objects. Remove bets to findout another (subset of) four. Continue removing fours untilnone is left," howmany times this can be done Ask: "How many subsets (groups of four) didyou remove?" and (b) understands that this Repeat to give the required number of test items,using sets of objects whose repeated subtrac- numbersare "evenly" divisible (i.e. no remainders), and inserting the appropriates tion can give an numbers, in instructionsand questions. answer to a

division question. Acceptable Response:

Thechild carriesout the actions and states the number correctly.

Continued NUMBER =1.mmnanwIM

TEST MATERIALS TEST PROCEDURES and COMMENTS.

(a) ii.A box of rods, (a) u,PROCEDURE:

Place before the child a rod, or rods, representing a number divisible by four, (Indicate that white is the unit.)

Say: "Make a crimson train for this." "Remove one trainrod, Remove another, Continue removing trainrods until none is left, How many did you remove?"

Repeat, to give .the required number of test items, using rods representing numbers which are "evenly" divisible, and inserting the appropriate colour words in instructions and questions,

Acceptable Response:

The child carries out the actions and states the number correctly,

, A collection of (b) i,PROCTURE: objects. When the child has carried out the procedures outlined in 33 (a)i, ask:

"How many fours in that number ?"

Repeat this form of questioning throughout the required number of test items.

Acceptable Response:

The child observes the subsets removed and states, for example,

"There are four fours in sixteen," (Continued over)

1 7 NUMBER

OBJECTIVES, TEST MATERIALS TEST PROCEDURES and COMMENTS, The Child

Imalolmo.MMEIMIAMINI 33, (continued) (b) ii,PROCEDURE;

When the child has carried out theprocedures outlined in 33 Wu, ask:

"How many fours in thatnumber?"

Repeat this form of questioning throughout the required number oftest items.

Acceptable Response:

The child observes the rods that havebeen removed and states, for example,

"There are four fours in sixteen,"

Note: It may be advisable in both (b)iand (b)ii to asfurther:

"How do you know?"

The child should reply, in hisowd words, to the effect that (for example) sine four fours can be subtracted from sixteen, there must be four fours insixteen, NUMBER

OBJECTIVES. TEST MATERIALS TEST PROCEDURES and COMMENTS.,

The Child.

34. Records, in number sentences using the signs +x=,the actions for addl tion, subtraction, multiplication, finding fractional parts and division,

Note:These actions

Were tested in

Objectives 21, 23, 24, 26,27,28,30,31,32 and 33, The items which follow (ix) test the recording of the actions.

Each test item should be presented at the appropriate stage in the learning process.

Records the com- i. Pencil and paper i. PROCEDURE: bining of two sets (addition) A number of sets of Sets: (Recording of objects with appropriate Place two of the sets before the child, Objective 21), numbers of members.

Say: "Combine these two sets."

"How many in the new set?"(This question may be omitted.) "Write the number sentence," (Continued over) NUMBER

OBJECTIVES. TEST MATERIALS TEST PROCEDURES and COMMENTS. The Child

34. (continued) Repeat, using other pairs ofsets, to give the required number oftest items,

Pencil andpaper Rods:

A box of rods. Place the box of rods before the child and give him two suitablerods, stating that white is the unit,

Say: "Place these rods end-to.end.Make a pattern."

"Read the pattern in colour." (Thisinstnl%: icyi may be omitted.) "Write the number sentence."

Repeat, using other pairs of rods,to give the required number of test items.

Acceptable Response:

The child performs the action andwrites the number sentence correctly,using the form 5 + 3= 8.

Note: If the child is uncertain,say "Tell me the number sentence first, Now write it down." If the child is thensuccessful, an intermediatestage has been reached.

. Records the ii.Pencil andpaper ii.PROCEDURE: removal of a

subset (sub Sets of objects. Sets: traction) (Recording of Place the set of objects before the child, Objective 23.)

Say: "Separate this set intotwo subsets and remove one subset."

h u Continued NUMBER

OBJECTIVES, TEST MATERIALS TEST PROCEDURES and COMMENTS. The Child...

34.(continued) Ask: "How many did you start with? How many did you remove? How many are left?" (These questions may be omitted.)

Say: "Write the number sentence."

Repeat, using other sets and subsets, to give the required timber of test items.

Pencil and paper Rods

A box of rods, Place a rod before the child, stating that white is the unit.

Say: "Make a pattern for this rod, using two more rods."

Ask: "What rods did you use? What numbers do they represent?" (These questions may be omitted.)

Say: "Place one of the shorter rods on top of the long rod," "Write the number sentence."

Repeat, to give the required number of test items, commencing with other rods and making other patterns, then placing one of the shorter rods on top of a long rod,

Acceptable Response:

The child performs the action and writes the number sentence correctly, using the form8 5=3,

Note: If the child is uncertain, say "Tell me the number sentence first. Now write it down,"If the child is then successful, an intermediate stage hasbeen reached. (Continued over NUMBER

OBJECTIVES, TEST MATERIALS TEST PROCEDURES and COMMENTS. The Child

34. (continued) iii.Records the com iii. Pencil and paper iii.PROCEDURE: paring of two sets (difference) A number of sets of Sets: (Recording of objects with different Objective 24.) numbers of members Place two sets of objects before the child, each set containinga different number of objects.

Say: "Pair the elements in thesesets to find the difference."

Ask: "Are they the same in number? What is the difference?" (Thesequestions may be omitted.)

Say: "Write the number sentence."

Repeat, using other pairs of sets, to give the required number oftest item!.

Pencil and paper Rods:

A box of rods. Place before the child two unequalrods. State that white is the unit.

Say: "Place these rods side byside, to find the difference."

Ask: "Are they the same? What is the difference?"(These questionsmay be omitted.)

Say: "Write the number sentence."

Repeat, using other pairs of rods,to give the required number oftest items.

Continued NUMBER

OBJECTIVES. TEST MATERIALS TEST PROCEDURES and COMMENTS. The Child

14. (continued) Acceptable Response:

The child performs the action and writes the number sentence correctly, using the form 10 9=1.,

Note:If the child is uncertain, say "Tellme the number sentence first. Now writeit down."If the child is then successful,an intermediate stage has been reached. v.Records the iv.Pencil and paper iv. PROCEDURE: deriving of sub-

traction facts Sets of objects. Sets from addition

facts. (Record Place before the child a set of objects. ing of Objective

26,) Say: "Separate this set into two subsets." "Write the number sentence." " "Now write as many other number sentences about thisas you can think of, both addition and subtraction."

Repeat, to give the required number of test items, startingon each occasion with a set containing a different number of members.

Pencil and paper Rods

A box of rods. Place before the child one of the longer rods, stating that white is the unit.

** The test proper begins here.

(Continuedover) NUMBER

OBJECTIVES. TEST MATERIALS TEST PROCEDURE P and COMMENTS. The Child

4. (continued) Say: "Make a pattern, with two rods in the secondline," "Write the number sentence."

"Now writeas many other number sentences from this pattern as youcan think of, both addition and subtraction."

Repeat, to give the required number of test items,using other rods and patterns.

Acceptable Response:

The child performs the action and makesan initial recording in the form 7 = 52. He follows this with other numbersentences in the form:

7=2+5 7-5=2 2=7-5 5+2=-7 7-2=5 7-5-2=0 2+5=7 5=7-2 7-2-5=0

The minimum requirement forsuccess is two addition and two subtraction forms.

Records thecorn- v. Pencil and paper v. PROCEDURE: billing of equiva

lent sets A collection of objects Sets (multiplication). from which equivalent (Recording of sets can be made, Place a number of equivalent sets before the child, (forexample, four sets, each Objective 27). with two members).

Say: "Combine these sets to makea new set." "Write the number sentence."

Repeat, to give the required number of test items,using other groups of equivalent sets. 11 1 NUMBER

TEST MATERIALS TEST PROCEDURESandCOMMENTS.

Pencil and paper Rods

A box of rods. Place before the child several rods of thesame colour.

Say: "Place these end.to.end andmeasure them." "Write the number sentence."

Repeat, to give the required number of test items, using other trains of rods.

Acuptable Response:

The child performs the action and writes the number sentence correctly,using the form

4 x 2 = 8 (meaning four twos equal eight)

If the child responds usinga repeated addition form, ask:

"Can you do it another way?"

Note: If the child is uncertain,say:"Tell in. the number sentence first. Now write it down,"If the child is then successful,an intermediate stage has been reached. vi.Pencil and paper vi. PROCEDURE:

A collection of objects. Sets

Place before the child a set of objects withan even number of members.

(Continued ow) NUMBER

Armem110

TEST MATERIALS TEST PROCEDURES and COMMENTS.

Say: "Separate this set into to subsets ofthe same number each."

Ask: "What part (fraction) of the set is this?"(indicating one subset),

Say: "Write the number sentence."or "Write the number sentence about that fraction."

Repeat, to give the required number oftest items, using other sets with aneven number of members.

Pencil andpaper Rods

A box of rods, Place before the child a rod representingan even number, stating that white is the unit.

Say: "Make a two-rod train for this rod."

Ask: "What part (fraction) of the train is/this rod?" (indicatingone of the train rods).

Say: "Write the number sentenu."or "Write the number sentence about that fraction."

Repeat, to give the required number oftest items, using other rods representing even numbers.

Acceptable Response:

The child performs the action and writes thenumber sentence correctly in the form

4= x x8 ar x8 :4 1 1 (j Continued NUMBER

OBJECTIVES, TEST MATERIALS TEST PROCEDURES and COMMENTS. The Child.

)4. (continued) Note: If the child is uncertain,say:"Tell me the numbersentence first. Now write it down."If theWild is then successful, an intermediatestage has been reached. di. Records the vii. Pencil and paper vii. PROCEDURE: finding of

thirds, quarters, A collection of objects Thinis fifths (Recording of A box of rods, Place before the childa set of objects, the number of which is divisible by three, or Objective 2 0.) a rod (or rods) representing a number divisible by three (white being the unit).

Say: "Separate this set into three subsets each withthe same number" (if working wish sets) or

"'lake a threerod train for this rod" (if workingwith rods), 4

Ask: "What part (fraction) is this?" (indicatinga subset, or a train rod).

Say: "Writea number sentence."

Repeat, to give the required number oftest items, using other appropriate sets or rods.

Acceptable Upon:

The child performs the action and writes the numbersentence correctly, in the form

3 = x 9 or 3X9=3

(Continued over) NUMBER

AIIII=111111101WWWWWwwM11.

TEST MATERIALS TEST PROCEDURES and COMMENTS.

Note: If the child is uncertain,say:"Tell re the number sentence first, Now write it downs"If the child is then successful,an inter.iediate stage has been reached,

Pencil and paper Quarters, fifths,

A collection of objects The procedure outlined for thirds is adapted for other fractions.

A box of rods. Place before the child sets of objectsor selected rods for numbers divisible by four, five,

Request the child to partition the set into four, five, ,parts, or to make for the selected rod(s) a 'train of four, five,...rods,

indicate one of the subset, or one of the train-rods, and ask what fractionit represents,

Ask the child to write the number sentence,

Repeat for each fraction, to give the required number of test items for each,

Acceptable Response:

The child performs the action and writes the number sentence correctlyin the form:

,it i x8 = 2 or 2 = X 8 4

1 n . 1 Ix 10 = 2 or ... Ix10 1 i 6x6 = 1 or 1r.Ix 6 NUMBER

OBJECTIVES. TEST MATERIALS TEST PROCEDURES and COMMENTS. The Child

34, (continued) If the child is uncertain, say:"Tell me the number sentence first. Now write it down." If the child is then successful,an intermediate stage has been reached. Note: Items viii, ix and x relate to forms of division. The division sign is not used in the early stages. Answers to division questions Ire recorded in the form of multiplication number sentences, viii, Records the viii. Pencil and paper viii, PROCEDURE: partitioning of

a set for sharing. A collection of objects. Sets (Recording of

Objective 31.) Place before the childa set of objects the number of members being divisible by a selected number, for example, by six,

Say: "Share these objects among six people; findout what each person's share will be." "Now write the number sentence."

Repeat, to give the required number oftest items, using other appropriate sets.

Pencil andpaper Rods

A box of rods. Place before the child a rod or rods representinga number divisible by a selected number, for example, by six,

(Continued over) NUMBER

TEST MATERIALS TEST PROCEDURES and COMMENTS.

1=wd.swi11==1MI=111~1MINPRIMA

Say: "Share this numberamong six people; find out what each person's share will be," "Now write the number sentence."

Repeat, to give the required number oftest items, using other appropriate rods.

Acceptable Response:

The child performs the appropriate action,partitioning the setor making a train, and records his findings ina number sentence in the following form:

12 = 6x 2 or 6 x 2 = 12

to indicate each "share" would consist of two things ("Twelveis six twos." or "There are six twos in twelve.")

Note: If the child is uncertain,say: "Tell me the number sentence first. Now write it down." If the child is then successful,an intermediate stage has been reached.

The child may find it helpful to writelhe number sentence firstas 12 = 6 x 0 and then fill in the frame.

If tite child writes 1x i2 = 2, indicating that each flare is a sixth

of the set, and that this istwo, he is not incorrect (in fact he showsa mature understanding).

Say: "Can you write it another way?"

1. Continued NUMBER

OBJECTIVES. TEST MATERIALS TEST PROCEDUR9S and COMMENTS. The Child

34. (continued) ix. Records the ix, Pencil and paper ix. PROCEDURE: partitioning of a set to find out Sets of objects. Sets how many equivalent sub Place before the childa set of objects, the number of members being divisible by sets there are fta selected number (three). (division) (Recording of Say: "Separate this set into subsets of three,to find out how many there are." Objective 32.) "Write the number sentence."

Repeat, to give the required number oftest items, using sets with the number of members being divisible by other selected numbers.

Pencil andpaper Rods

A box of rods. Place before the child a rod, or rods, representing a number divisible bya selected fi f number (three). State that white is the unit.

Say: "Make a train to fmdout how many threes are in that number," "Write the number sentence."

Repeat, to give the required number oftest items, using rods representing numbers divisible by other selected numbers,

Acceptable Response:

The child performs the action andwrites the number sentence in the form:

4 x 3 = 12 or 12 = 4 x 3

ttThe number may be varied, (Continued over)

121 NUMBER

=104=walim1111.

OBJECTIVES. TEST MATERIALS TEST PROCEDURES and COMMENTS. The Child Mir NINNOMwm1111111,1.....

34,(continued) to indicate that thel are four threes in twelve, or that twelve consists of four threes,

Note: If the child is uncertain,say: "Tell me the number sentence first, Nowl write it down," If he is then successful, an intermediate stage has been reached.

The child many find it helpful to write the number sentence first ash

12 r. 0 x 3 and then fi ll in the frame.

Records the x.Pencil and paper x.PROCEDURE: removal of

equivalent sub. Sets of objects. Sets sets. (Recordinc

of Objective 33,) Place before the child a set of objects the number being such that theset canbe partitioned into equivalent subsets, For example, a set whose number is divisibl tt by four would be suitable; the piocedure set out below is based on sucha set,

Say: "Remove four objects; remove another four; continue removing foursunt none is left."

Ask: "How many fours did (could) you remove?" (TI,;s question may be omitted.)

Say: "Write a number sentence which tells howmany fours in that number,"

Repeat, to give the required number of test items, selecting other appropriat numbers of objects. MI.1.

ttThe number may be varied, Continued over NUMBER

TEST MATERIALS TEST PROCEDURES and COMMENTS.

Pencil and paper Rods

A box of rods. Place before the childa rod or line of rods representing a number for which a train

can be made. (Indicate that white is the unit.) For example, rods for whicha ftcrimson train can be made would be suitable; the procedureset out below is based on such a line of rods.

Say: "Make a crimson train for this. Tellme which number each trainrod represents. Remove a four, remove another four; continue removing fours until none is left."

Ask: "How many fours did (could)you remove?" (This question may be omitted.)

'ay: "Write a number sentence which tells howmany fours in that number."

Repeat, to give the required number of test items, selecting otherappropriate rods and numbers.

Acceptable Response:

The child carries out the actions and writes the number sentencecorrectly, in the form:

3x4 =12

which indicates that thereare three fours in twelve.

Note: If the child is uncertain,say:"Tell me the number sentence first.NOS write it down."If the child is then successful,an intermediate stage has been reached.

tt The colour may be varied, NUMBER

IMIIEMMIIMMORIMM

OBJECTIVES. TEST MATERIALS TESTPROCEDURES and COMMENTS.

Thi ChM.

35. Understands the Using Sets of Objects: value of the

written symbols I. A collection of objectscon i. Recording given numbers representing sisting of ten bundles of

numbers beyond ten, and ten single objects PROCEDURE: ten, to ninety. e.g. the bundles may nine. consist of used matches Place before the child three bundles of ten and four single objects.Ensure that the (Numeration) held together with child knows that there are ten in each bundle, rubber bands.

Ask: "How many 'things' are there?"

Say: "Write that number."

Ask: "Why did you write 3?" "Why didyou write 4?"

Repeat with other numbers of bundles and single objects,to give the required number of test items.

Acceptable Response:

The child states the number correctly, "thirtyfour" (not "threetens and four") and writes the numeral 34. He states in hisown words that the 3 refers to the three bundles of ten, and the 4 refers to the four Singe objects.

H.Cards each with a numeral ii.Interpreting given numerals of value greater than ten

(to 99) PROCEDURE:

Objects as in 35 i. above. Show the child a card witha numeral, for example 34.

124 Continued NUMBER.

OBJECTIVES, TEST MATERIALS TEST PROCEDURES and COMMENTS, The Child IIMPMIM.PIMONEMOM11

35, (continued) Say: "Make this number,"or "Make a set with this number."

Ask: "How many things did you use?""Why did you get three bundles of ten?" "How many is that?""How did you know to get four 'singles'?"

Repeat, showing other numeral cards, to make the required number of test items.

Acceptable Response:

The child makes the required number, using tens and units, and explains thereasons for his selection. For example, with the card for 34, he states in hisown words that the first numeral, 3, represents the three bundles of ten (and that this is thirty), tad the last numeral, 4, represents four units.

Using Rods

1. A box of rods. . Reeding given numbers

PROCEDURE:

Place before the child three orange rods and a crimson rod. Indicate that white is the unit.

Ask: "What number is that?"or "What number do these rods represent?"

Say: "Write the number."

Ask: "Why did you write 3?" "Why did you write 4?" (Continued over NUMBER

OBJECTIVES. TEST MATERIALS TEST PROCEDURES and COMMENTS. The Child

35. (continued) Repeat with other rods, to make the required numberof test items.

Acceptable Revue:

The child states the number correctly, for example, "thirty.four"(not "tree tens and four") and writes the numeral 34. Hestates in his own words that the 3 refers to the three tens (the orange rods), and the 4 refersto the four units (crimson rod).

ii.Cards each with a numeral 1 Interpreting givennumerah. of value greater than ten.

(to 99). PROCEDURE:

Show the child a card witha numeral, for example, 34. indicate that white is the unit.

Say: "Make this number with rods."

Ask: "What number have you made with the rods?"

"Why did you get three orange rods? What number is that?Why did you get a crimson rod?"

Repeat, showing other numeral cards, to make the requirednumber of test items.

Acceptable Response:

The child makes the required number, usingorange rods with another rod, He states in his own words that, for exrniple, the first numeral, 3,represents the three orange rods, which is thirty, and that the last numeral, 4,represents the crimson tod, which is four units, NUMBER

OBJECTIVES. TEST MATERIALS TEST PROCEDURES and COMMENTS. The Child...

36, Knows how to Pencil and paper. PROCEDURE: write the

numerals for Dictate to the child ten numbers one ata time: Seventeen, twenty-four, thirty-two given numbers forty-six, fifty.one, sixty-three, seventy-eight, eighty-nine, ninety, twelve. beyond ten,

to 99 Say first: "Write (numerals for) thesenr ." (Notation).

AcceptableResponse:

The child writes 17, 24, 32, 46, 51, 63, 78, 89, 90, 12, correctly. The childmum succeed in all ten items. It is not necessary that the child indicate place-value by writini in vertical columns.

37, Uses memorised Each set of test items should be presentedat the appropriate stage of the learning process. number facts

in number These tests of memorisationare to be given only after sufficient experience of written work using concrete material sentences,

Cards, or PROCEDURE: Duplicated sheets, or

Chalkboard copy, One set of examples only should be presented ata time, and corrected before proceed each showing a set ing. of five examples

in one of the cate For each category,one suitable set of five examples is shown, If a further set is requiret gories set out below the teacher should devise five more examples similar to those shown. Insome cases Pencil may be necessary to repeat examples to obtain the required number in the seconc Blank paper (if using set.

chalkboard copy). (Continued over: NUMBER

OBJECTIVES. TEST MATERIALS TEST PROCEDURES and COMMENTS. The Child

37. (continued) ,,-tow the child the set of examples. Explain whether theanswer is to be inserted, or the whole number sentence copied. Concrete material is not to be used for this Objective.

Acceptable Response:

For each category, the child shouldanswer correctly five examples out of five (i.e. none wrong),or at least eight out of ten (i.e. only two wrong allowed).

In a few cases (e.g. for testingquarters, within the range of ten) it will not be possible to devise sufficient differenttest questions. The teachermay repeat some of the questions. Alternatively,a reduced number may be presented. In such cases the child shouldanswer correctly all the examples which are given.

gte: After written tests, interviewtype follow-upmay reveal further informa- 4q.ut a child'sresponses.

Addition categories Set of example- each category lived:

Total to 10. No Zero

1,Two addends, larger first, 6+3:0 7+1=0 5+4[1 6+2:0 frame for total last,

2. Two addends, smaller first, 2.3+6=-0 2+5.--.0 1+8:0 4+6=0 2+00 frame for total last.

Three addends, frame for 3.2+4+1:0 1+2 +3:0 6+2 +1 :0 3+1+3:0 4+2+3g0 total last.

128 TEST MATERIALS TEST PROCEDURES and COMMENTS.

.16,1.1P 4. Frame for total first; two 4, =6+3 =6+2 04+1 411.111. 0=4+3 0=5+4 addends, larger first.

$1MIIIMP 5. Frame for total first; two 5, 0=1+6 =2+3 111=3+4 0:4+5 0=3+7 addends, smaller first,

6, Frame for total first; 6. =3+5+1 El:2+4+2 4.11i111 0 =1+8+1 0=2+2+3 three addends.

7, Two addends, frame for 7, 4+ =7 5+ri=8 9+ =10 2+ second; total last,

8. Two addends, frame for 8. 0 +2=6 +1=8 0+4=9 Q+3=6 +2=10 first; total last.

9, Three addends, frame for 9. 2+4+ 0 1+2+ 0 =6 5+3+0=10 3+1+0= 2 +6 +Q=9 third; total last,

10, Two addends equal two 10, 6+1=3+1] 3+2=4+ El 5+4=7+ El 4+2=1+ 3+5=5+ 1] addends, frame for last,

Total to 10 (including zero),

11, Zero as one addend, 11. 8+0 =8 7+0=7 5+ 0 =6 3+0r.3 9+ occuring second, in frame; (includes two examples from category 1, to prevent perseveration) total last.

(Continued over) NUMBER

OBJECTIVES. TEST MATERIALS TEST PROCEDURES and COMMENTS. The Child .A.....immbb.1111FmMl

37, (continueill Total to 20 (no zero).

12, Frame for total last; two 12.9 +3= 6+5= El 9+7= 8+S= 7+4= addends each less than ten, larger first.

.1=01=11V 13, Frame for total last; two 13, 5+8: 4+7=10 3+9=n 5+9=0 addends each less than ten, smaller first.

14. Frame for total last; two 14. 12+4= Q 13+5= 17+2= 0 16+4= Q 11+8= 0 addends, one over ten, larger first.

15. Frame for total last; two 15.5 +13 = 0 2+17=El 6+11= 5+14 = n 6+12= addends, one over ten, smaller first,

16. Frame for total last; three 16.5 +7 +2= 0 7+7+1= 0 2+5+9= 6+9+3= Q L1 8+1+8: addends, each below ten.

17, Frame for total last; three 17.2 +4 +12 = 13+3+3=0 1+15+2= 0 11+2+6= Q 3+5+12= 0 addends, one over ten.

18. Frame for total first; two 18. =7+4 =9+3 =8+6 0 =9+7 0=7+6 addends below ten, larger first.

19, Frame for total first; two addends below ten, smaller first. 130 Continued NUMBER

TEST MATERIALS TEST PROCEDURES and COMMENTS.

IMMINE,

20. Frame for total first; two 20. fl =13+3 0 =15+2 0=11+7 7=13+5 I =12+8 addends, one over ten; larger first.

21. Frame for total first; two 21.0 =4+15 =3+16 0 =5+12 =9+11 0 =1+14 addends one over ten; smaller first,

22. Frame for total first; 22. =2+6+5 El=3+9+4 0 =8+6+5 =3+8+7 =7+5+8 three addends all below ten.

23. Frame for total first; 23. 0 =3+14+2 =2+16+2 0 =15+3+1 =4+3+12 Q =4+2+1: three addends, one over ten.

24. Total last; two addends 24, 8+ =12 9+ 0 =15 9+0=18 7+ 0 =.13 8+ =16 below ten; frame for second.

25. Total last; two addends 25,0 +9=14 +6=11 0+7=15 +9=18 +8=17 below ten; frame for first.

26. Total last; two addends, 3 +E 26, 2+0 =18 =15 6+ 0 =20 3+ 0 =19 1+0=17 one over ten; frame for larger number in middle.

27. Total last; two addends, 27. 0 +14=19 +15=16 +13=17 0 +12=15 0 +19=20 one over ten; frame for smaller number first.

(Continued over) NUMBER

OBJECTIVES, TEST' MATERIALS TEST PROCEDURES and COMMENTS, The Child 111 37. (continued) 28, Total last; three addends 28. 6+ +3=15 7+2+^=16 El +5+8=20 8+ +1=18El+5+5=15_ below ten, frame for any,

r_, 29. Total last; three addends, 29. 3+12+ =202+11+ Li =18 +15+2=19 14+2+ =17 +3+141.c one over ten; frame for one of the smaller addends.

Total to 20, zero included.

30. Total last; zero as one 30, 19+ ri =19 18+ = =18 13+ 11=15 12+ =12 17+ E] =20 addend; frame for zero, (includes two examples fromcategory 14, to prevent perseveration)

31, Two addends eqiial two 31. 8+7=9+ 0 7+6=10+ fl 12+8=20+ 9+8=8+ 6+5=4+ addends; frame for last.

Subtraction

Note: In the number sentence 6 4 = 2 6 is called the minuend 4 is called the subtrahend 2 is called the difference

Minuend to 10. No zero,

1, Difference frame last; subtrahend, 1 or 2.

Continued NUMlitti

OBJECTIVES, TEST MATERIALS TEST PROCEDURES and COMMENTS. The Child

37, (continued) 2. Difference frame last; subtrahend larger than difference.

3. Answer frame last; two 3. 9-2-1= 10-5-3= 8-4-2= -7-2-4= Q 10-1-6= E subtrahends.

4. Frame for difference first; 4. =9-1 =10-3 at =8-3 =9-4 subtrahend smaller than difference,

5. Frame for difference first; 5. =10-7 Q =8-6 =9-5 =7-5 =6-4 subtrahend greater than difference,

6. Frame foranswer first; =7-2-1 Q- ,0-3-4 =9-4-2 Li =8-1-6 two subtrahends.

Minuend to 10, Zero included.

7. Difference frame last; zero is subtrahend.

8, Difference frame last;zero 8, 6-6=0 10_10. 7_6= El 8-8= 9-1= is difference in frame. (contaii t two examples from categories ! and 2 to prevent perseveration)

41.111, 1.11nor 9. Difference last, frame for 7 9. 6 =4 10 j 8 El =5 subtrahend; subtrahend smaller than difference.

(Continuedover) NUMBER

PIM OBJECTIVES, TEST MATERIALS TEST PROCEDURES and COMMENTS, The Child

37. (continued) 10, Difference last, frame for subtrahend; subtrahend is greater than difference,

11. Difference last; difference 11. 6 =0 10 =0 5 TL: =0 9 rzo is zero; frame for subtrahend,

12. Difference last, frame for 12. 6 =6 10 =10 6 =5 8 0 =3 9 :9 subtrahend. Subtrahend is zero. (contains two examples fromcategories 1 and 2 to prevent perseveration)

Minuend to 20. Nozero.

13. Frame for difference last; 13. 15-1= 17-1= 16-1= 184= J 13-2= subtrahend is 1 or 2.

14. Frame for difference last; 14. 15-9= 13-7= 0 14-8= 12-7= 174= subtrahend is greater than difference but under 10.

15. Frame for difference last; 15. 15-12= 18-14= 0 n ta= 0 20-15= 19-11= subtrahend is larger than 10.

16. Frame for answer last;two 16. 16-2-3= 0 17-1-4= 12-2-3= 0 20-8-7= [i subtrahends both under 10.

17. Frame for answer last;two 17, 16-12-2= 14-11-1= 20-2-14= a 13-1-11= a 19-13-5g subtrahends, one over 10.

1 31 Continued NUMBER

OBJECTIVES. TEST MATERIALS TEST PROCEDURE;; and COMMENTS. The Child

37, (continued) Minuend to 20. Zero included,

18. Frame for difference last; 18, 16CP 0 174 El 204 Q 134 11 114 zero is subtrahend,

19, Frame for difference last; 19. 16-16=El 18-18=U 17-2= El 15-1= 0 17-17= El zero is difference in frame, (contains two examples from category 13 to prevent perseveration)

20. Frame for difference first; 20, [i =15-1 =13-2 =17-1 Li =18-2 =19-1 subtrahend is 1 or 2.

21. Frame for difference first; 21. =15-9 =14-8 El =13-7 Q=12-8 El =11-7 subtrahend is greater than difference, but under 10.

22. Frame for difference first; 22. [1 =15-12 a =11 -13 E] =19 I1 =20-14 0 =16-12 subtrahend is greater than

10.

23, Frame for answer first; 23. Q =216-2-3 =18-5-4 Q =12-2-3El =20-7-6 Q =18-2-9 two subtrahends, both under 10,

24. Frame for answer first; two 24, El =16-12-211] =18-15-1 Q=11 -2 -14 =18-3-12 0=20-4-1 subtrahends, one over 10,

25. Difference last; frame for 25. 16 a =14 17 El =13 18 El =9 14 =7 20 a =I 1 subtrahend; subtrahend under 10.

(Continued over) NUMBER

OBJECTIVES. TEST MATERIALS The Child ... TEST PROCEDURES and COMMENTS.

37. (continued) 26. Difference last; f;a;se for 26. 16 =2 18 =1 20-- =4 12 Li =1 subtrahend; subtrahend 17 0 =2 over 10.

27. Difference last; frame for 27. 16 0 =0 17 =0 13 =0 20- 0 =0 subtrahend; difference is 11 0 =0 zero. 28. Difference last; frame for 28. 16 0 =16 8 0 =18 13 0 =12 19 0 =18 subtrahend; subtrahend 12 0 =12 is zero.

Multiplication

Product to 10. Two factors, no zero.

1. Factors first; frame for 1. 2x3= Ei 4x2= 0 3x3= 0 6x1= product. 5x2= 0

2. Product first; frame for 2. 0 =2x3 0 =3x? =7x1 =2x4 product. =5x2

3. Product last; frame for 3. 0 x3=6 0 x2=8 0 x5=10 0 x4=8 first factor. El x6=6 (Type: How many threes in six?)

1:3G Continued NUMBER

TEST MATERIALS TEST PROCEDURES and COMMENTS.

4, Product last; frame for 4, 2x E -=6 3x =9 5x r10 4x =4 second factor.

(Type: Two "whats"are six?)

S, Product first; frame for 5, 6= x3 8= x4 10-rix5 9= 0 x3 first factor. (Type: Six is howmany threes?)

MEM, 6, Product first; frame for 6, 6=2x 8=4A 9=3x [-.] 7=7x 10=5x second factor. (Type: Six is two of what?)

Product to 20. Two factors. No zero, All factors under 10,

7. Product last; frame for 7, 5x3= 4x5= Q 8x2: 0 2x9: 4x4= product,

8, Product first; frame for 8, =5x3 Q =2x8 0 =7x2 =3x4 a=6x3 product.

9, Product last; frame for 9. Q x3=15 x4=16 x2 :12 L1 x6=18 El x4:12 first factor, (Type: How many threes in fifteen?)

10. Product last; flame for 10. 5x 1:1 =15 4x =12 3x ti =18 4x E =16 2x [1 =14 second factor,

(Type: Five "whats" are (Continued over) fifteen?) NUMBER

OBJECTIVES. TEST MATERIALS TEST PROCEDURES and COMMENTS. The Child

37. (continued) I L Product first; frame for IL 15= x3 18= x6 14= x7 21 x5 12= 0x2 first factor.

(Type: Fifteen, howmany threes?)

12. Product first; frame for 12. 15=5x 2f:4x El 12=3x 18=3x Q 16=4x second factor, (Type: Fifteen equals five of what?) (Used in "Share 15 lollies among 5 children.")

Fractions,

Range to 10.

1 1. Halves; frame last for 1. -2X1( -1.21(8= -2x4f.= j" smaller number,

2. Halves; frame last for 2.5 -2x 2=4 11-1X 2 2 larger number.

ri 1 71 3, Halves; frame in middle 3.2x =5 -X Li=4 -X Li=I 2X 2 2 for larger number.

4. Halves; frame first for 4, smaller number. NUMBER

TEST MATERIALS TEST PROCEDURES and COMMENTS. m....010.0=.=NIMM wimM 11111 5.Quarters; frame last for 5. 48= [1] 1x4= 1x8= ix4= 1-x8= 4 4 4 4 4 smaller number. - (Repetition of examples is necessary to makeup numbers)

...... 6. Quarters; frame last for 6. 24x 2=1X 1dX 24X 4 _. 4 ww 0 larger number, (see note above)

7. Quarters; frame in middle 7. -1 x =2 -41x J,fx 0 =2 _Ixiirl=1 ix [11:2 4 g.111111 4 4 for larger number (see note above)

, 8. Quarters; frame first for 8. =48 =x8 4 4 41 smaller number. (see note above)

9. Thirds; frame first for lx6= ix= 118111 3 3 smaller number. (see note above)

_1 10. Thirds; frame last for 10. 3=-}x 1- !M. 3 .11111M larger number. (see note above)

ix 11. Thirds; frame in middle 1-x 1=1 -3X 3 3 - for larger number. (see :..ate above)

1 12. Thirds; frame first for 12. El "zi =-x9 =.1-x6 111011110.11 =-3X9 3 7 smaller number. (see note above)

Note:Similar tests, if desired, may be devised for a fifth of, a sixth of, a seventh of, an eighth of, a ninth of, and a tenth of, based on: (Continued over)

13 NUMBER

OBJECTIVES. TEST MATERIALS TEST PROCEDURES and COMMENTS. The Child.

37. (continued)

Range to 20,

13. Halves; frame last for 13. ix2C0 xl6= 2 2 smaller number. 2 LJ

14. Halves; frame last for 14. 1C4x 9=.1-2x 74x larger number.

15. Halves; frame in middle ri 15. 2x El =10 XLI =7 2 Li =9 for larger number. 2x

16. Halves; frame first for 16, 0 zl4x20 1:1?}x14 =-1-2-x18 =12-x12 E =2x10 smaller number.

17. Quarters; frame last for 17. i.x16= 4x12.= a 4x20=Q lifx16= Q 4x20 smaller number. (Repetition of examples isnecessary to make up number) 18. Quarters; frame last for 18. 44x 34x 0 54xU 44x 54x 0 larger number. (see note above) 1 4 Continued NUMBER

TEST MATERIALS TEST PROCEDURES and COMMENTS.

1 =3 19. Quarters; frame in middle 19. 4x r.4 -4 X 4x11 =S 4xx .74 .11./0,11 11. for larger number, (see note above)

20. Quarters; frame first for 20. f =ix16 :-.1x12 =ix16 Ax20 4 4 smaller number. (see note above)

1 21. Thirds; frame last for 21, 3 x15= ix 1 2= x18.:-. x15= X18=1-1 3 .-. 3 3 Alw 3 LI smaller number, (see note above)

22. Thirds; frame last for 22. 5=131 67-.x 0 5=4 [1] @x 6dx 3 3 3 3 larger number, (see note above)

11mM.. 23. Thirds; frame in middle 23.3x =5 xL11-1.16 Ix =4 X 11=5 3 3 3 for larger number. (see note above)

24. Thirds; frame first for 24. 11 =3x15 =-isx18 11=3x12 .=-1-3 x15 smaller number. (see note above)

Note: Similar tests, if desired, may be devised for a fifth of, a sixth of, a seventh of, an eighth of, a ninth of, and a tenth of, basedon:

x 20 = 4 -018:3

x 14 r. 2

x16:72 (x18 =2 ix 20=2 NUMBER

OBJECTIVES. TEST MATERIALS TEST PROCEDURES and COMMENTS. The Child...

38, Applies the Concrete material with knowledge gained which to make structures and the number representing number sentences. facts learnt to composing envir °mental narra tives for the operations of (a) addition, (b) subtraction, (c) multiplication, (d) finding frac tional parts and (e) division.

(a) Addition (a) i,A collection of objects, (a) i. PROCEDURE:

Show the child two sets of objects, thencarry out the action of combining them to make a new set. For example,a set of five and a set of two may be pushed together to showa new set of seven.

Say: "Tell mea story about that." (a narrative, not a number sentence)

Repeat, using otherstructures, with actions, illustrating addition number seri tences (two addends), to make the required number oftest items.

Acceptable Response:

Using all the given numbers the childcomposes, and states orally, a narrative which shows that he applies hisknowledge of addition toan environmental. situation. For example: "I had five marbles, Iwon two more, then I had seven." NUMBER

OBJECTIVES. TEST MATERIALS TEST PROCEDURES and COMMENTS. The Child

38. (continued) (a) ii. Abox of rods. (a) ILPROCEDURE:

Show the child a pattern in which two rods endtund are measured with single rod, e.g. yellow and red measured with black, representing the number sentence S + 2 = 7.

Say: "Tell me a story about that pattern." (a narrative, not a number sentence)

Repeat using,other patterns illustrating addition number sentences (two add ends), to nae the required number of test items.

Acceptable Response:

Using all the given numbers the child composes, and states orally,a narrativi which shows that he applies his knowledge cf addition to an environmenta situation, For example: "My sister has five cents, my brother has two cents, Sc they have. seven cents altogether."

(a) iii. Cards, on each of which (a) iii.PROCEDURE: is one addition number sentence in the form: Show the child one of the cards,

5+2:7 Say: "Tell me a story about this number sentence."

Repeat, using other cards, to give the required number of test items.

Acceptable Response:

Using all the given numbers the child composes, and states orally, a narrativi which shows that he applies his knowledge of addition to an environmenta situation. For example: "There are five flowers on this plant and two flower on that. So that makes seven flowers,"

(Continuedaver NUMBER ,111.Nm OBJECTIVES. TEST MATERIALS TEST PROCEDURES and COMMENTS. The Child

38. (continued)

(b) Subtraction (b) i. A collection of objects. (b) i.PROCEDURE: (removal)

Using objects, show the childa structure, with actions, representinga numbei sentence, for example: 7 2 = 5 (i.e.a subset of two is removed froma set o: seven, leaving a subset of five.)

Say: "Tell me a story about that." (a narrative, nota number sentence)

Repeat, using other structures, with actions illustrating subtractikuy it 111041 to give the required number of test items.

Acceptable Response:

Using all the numbers given, thechild composes, andstates orally, a narrative which shows ,that he applies hisknowledge of "removal"subtraction to an environmental situation. For example: "Sevenbirds were on a tree. Two flew away, So five were left."

ii. A box of rods. (b) ii. PROCEDURE:

Show the childa rod structure in which a shorter rod is placedon top of another and the "gap" measured. For example,red on top of black, with yellow measure ing the "gap", representing the numbersentence 72 = 5,

Say: "Tellme a story about that." (a narrative, nota number sentence)

Repeat, using other structures with actions illustrating subtraction byremoval, to give the required number of test items.

Continued NUMBER

TEST MATERIALS TEST PROCEDURES and COMMENTS.

Acceptable Response:

Using all the numbers given, the childcomposes, and states orally, a narrative which shows that he applies his knowledge of "removal"subtraction to an environmental situation, For example: "Tony hadseven apples. He gave two away. So he had five left."

(b)ui. Cards, on each of which (b) iii. PROCEDURE: is one subtraction number sentence in the Show the child one of the cards. form: Say: "Tell me a story about this number sentence." 7 2 = 5

Repeat, using other cards showing subtraction numbersentences, to give the required number of test items.

Acceptable Response:

Using all the numbers Oven, the childcomposes, and states orally, a narrative which shows that he applies his knowledge of "removal"subtraction to an environmental situation. For example: "Mother madeseven patty.cakes. She gave me two, so that left five."

Place two sets of objects before the child and then pair themthus:

0.0 (for 75 = 2) 00 0-0 0.0 04 0 0 (the comparison aspectof,subtraction)

(Continued over) NUMBER

OBJECTIVES. TEST MATERIALS TEST PROCEDURES and COMMENTS. The Child ',...=6.mromar...... m

38, (continued) Say: "Tell me a story about this,"

Repeat, using other structures, with actions illustrating subtractionby coin parison, to give the required number of test items.

AcceptableReims:

Using all the given numbers, the childcomposes, and states orally, a narrative which shows that he applies his knowledge of "comparison"subtraction to an environmental situation. For example, "Iam seven and May is five. I am two years older than May." (or, "The difference in our ages is two years.")

Show the child a black and a yellow rod side by side, endslevel. Measure the difference with a red rod,

Say: "Tell me a story about that," (a narrative, nota number sentence)

Repeat, using other structures, with actions illustrating subtractionby come parison, to give the required number of test items.

Acceptable Response:

Using all the given numbers, the childcomposes, and states orally, a narrative which shows that he applies his knowledge of "comparison"subtraction to an environmental situation. For example, "I hit the bullaeyeseven times and Tom hit it five times, I scored twomore than Tom." (or, "The difference between our scores is two.") NUMBER

OBJECTIVES. TEST MATERIALS TEST PROCEDURES and COMMENTS. The Child

38. (continued) (c) iii, Cardson each of which (c) iii. PROCEDURE: is a number statement in the form: Show the child one of the cards (read it aloud if necessary),

"The difference Say: "Tellme a story about that." (a narrative, not a number sentence) between 7 and 5 is 2," Repeat, using other cards, with statements illustrating subtraction bycont parison, to give the required number of test items.

Acceptable Response:

Using all the given numbers, the childcomposes and states orally a narrative

wh!ch shows that he applies his knowledge of "comparison" subtraction toan environmental situation. For example, "Dick sleptseven hours and Harry slept five. Dick slept two hours longer than Harry."

(d) Multiplication (d) i,A collection of objects. (d) i,PROCEDURE:

Place before the child three sets of objects each with four members. Pushthem together to make a new set,

Say: "Tellme a story about that." (a narrative, not a number sentence)

Repeat, using other arrangements, with actions, illustrating multiplication,to give the required number of test items,

Acceptable Response:

Using the given numbers, the child composes, and states orally,a narrative which shows that he applies his knowledge of multiplicationto an environmental

(Continued over) NUMBER

TEST MATERIALS TEST PROCEDURES and COMMENTS,

situation. For example, "Thereare three boxes with four pencils in each, sc there are twelve pencils."

If the child composesa narrative in the repeated addition form, e.g. "There are four pencils in one box, four in aiother box andfour pencils in another box that is twelve pencils,"say "Can you tell me that, in another way?" to ascertain whether the childcan re-state the narrative in multiplication form.

(d) fi, A box of rods, (d) ii, PROCEDURE:

Place before the child three crimson rods. Put themencl.to.end and measure them with an orange anda red rod,

Say: "Tell me a story about that pattern." (a nand*,not a number sentence)

Repeat, using other train patterns, with actions,illustrating multiplication, tc Ole the required number of test items.

Acceptable Response:

Using the given numbers, the childcomposes, and states orally, a narrative which shows that he applies his knowledge of multiplicationto an environmental situation. For example, "Thereare three boxes with four pencils in each, so there are twelve pencils."

Note:If the childcomposes a narrative in the repeated addition form, for example "There are four pencils inone box, four in another box and four Irs another box; that is twelve 'pencils",say "Can you tell me that in another way?" to ascertain whether the child can restate the narrative in multiplication form.

Continued NUMBER

OBJECTIVES. TEST MATERIALS TEST PROCEDURES and COMMENTS.

The Child .

(d) iii, Cards on each of which (d) iii,PROCEDURE: is one multiplication number sentence in the Show the child one of the cards, form: Say: "Tell me a story about that." 3x4:12 Repeat,usingother cards, with sentencesillustrating multiplication, to give the required number of test items.

Acceptable Response:

Using the given numbers, the child composes, and states orally,a narrative which shows that he applies his knowledge of multiplication toan environmental situation. For example; "There are threecages with four birds in each, so there are twelve birds."

(e) i,A collection of objects. (e) i.With sets of objects

Halves:

A number of "dividers" PROCEDURE: for partitioning sets.

A "divider" may be a Place before the child a set which contains eight members. Usinga "divider", thin ruler, a pencil, a partition the set so that there are four members on each side. knitting needle or similar object.

(Continued over) NUMBER

TEST MATERIALS TEST PROCEDURES and COMMENTS.

Say: "I have cut the set in half. Tellme a story about that,"(a narrative, not a number sentence)

Repeat, halving other sets containingan even number of members, to give the required number of test items.

Acceptable Response:

Using the given numbers the childcomposes and states orally a narrative which shows that he applies a knowledge of halvingto an environmental situation. For example: "I had eight marbles,lost half of them,so I only had four left."

Note:The child bases his narrativeon the number sentence derived from the model, rather than on the model itself,

Other Fractions:

PROCEDURE:

Adapt the procedures above, partitioningappropriate sets into three, or four, .,subsets, to give examples involving thirds,quarters,.

Continued NUMBER

TEST MATERIALS TEST PROCEDURES and COMMENTS.

Repeat for each fractal, to give the requirednumber of test items for each.

Acceptable Response;

Using the given numbers the child composes, and statesorally, narratives which show that he applies a knowledge of thirds, quarters, , ... , to anenvironmental

situation.

(e) ii.A box of rods, (e) ii.With rods.

Halves:

PROCEDURE:

Place before the child a brown rod. Place two crimsonrods beside it

Say: "I have found half of that rod. Tell me a story aboutthat." (a narrative, not a number sentence)

Repeat, selecting other rods representing numberswhich can be halved, to give the required number of test items.

Acceptabk Rupontt:

Using the given numbers the child composes and statesorally, a narrative which shows that he applies a knowledge of halving to anenvironmental situation. Foi example; "I had eight lollies. I ate half of them, so Ionly had four left."

(Continued over NUMBER

OBJECTIVES. TEST MATERIALS TEST PROCEDURES andCOMMENTS. The Child

38. (continued) Other Fractions:

PROCEDURE:

Adapt the above procedures outlined for halves, usingappropriate rods, to give examples involving thirds,quarters,

Repeat for each fraction, to give the required number oftest items for each.

Acceptable Response:

Using the given numbers the child composes, andstates orally, narratives which show that he applies a knowledge of thirds, quarters, ,to an environmental situation,

(e) iii. Cardson each of (e) iii, With cuts. which isone fraction

number sentence, in Halves: the form

PROCEDURE: xl2=6 Show the childone of the cards.

Say: "Tellme a story about that number sentence."

Repeat, using other cards,to give the required number oftest items.

Acceptable Response:

Using the given numbers thechild composes, andstates orally, a narrative which shows that he appliesa knowledge of halving to an environmental situation. For example: "Tom had twelve pencils.He gave away half of them,so he only had six left,"

Continued NUMBER

OBJECTIVES. TEST MATERIALS TEST PROCEDURES and COMMENTS. The Child

38. (continued) Other Fractions:

PROCEDURE:

Adapt the above procedure using appropriate cards,to give examples involvini thirds, quarters,..

Repeat for each fraction, to give the required numberof test items for each Alternatively, includea variety of fractions in the set of test items,

Acceptable Response:

Using the given numbers the childcomposes, and states orally, narratives which

show that he appliesa knowledge of thirds, quarters,. , ,to an environmenta: situation. 6 (I) Sharing (1) i.A collection of objects. PROCEDURE: (division)

Place before the childa set of objects which contains twelve members. Divide the set into four equal "shares",

Say: "I have 'shared out' thisset of twelve into four shares. Tell mea story about that." (a narrative, nota number sentence)

Repeat, using other appropriatesets and other numbers of "shares", to give the required number of test items.

Acceptable Response:

Using the given numbers the childcomposes, and states orally, a narrative which shows that he applies his knowledge of sharingto an environmental dtuation,

(Continued over) NUMBER

TEST MATERIALS TEST PROCEDURES and COMMENTS.

For example: "Mother shared twelve lollies amongst her fourchildren; each received three lollies."

(f) h.A box of rods. (f) I PROCEDURE:

Place before the childan orange and a red rod, endtoend. Measure this with four lightreen rods.

Say: "I have divided the orange and red into four equal 'shares'. Tellme a story about that."

Repeat, using other appropriate rods, and other numbers of "shares",to give the requiid number of test items.

Acceptable Response:

Using the given numbers the childcomposes, and states orally, a narrative which shows that he applies his knowledge of sharing toan environmental situation.

For example: "I shared twelve marbles amongstmy four friends; each received three marbles,"

(f) ill. A number of cardson (f) iii. PROCEDURE: each of which is one sharing number Show the child one of the cards. sentence in the form Say: "Tell me a story about this sharing number sentence." 12 = 4 x 3 (four of three)

Repeat, using other cards, to give the required number of test items. (i.e. four "shares" of three each.) 5 Continued NUMBER ImivalmlterlMambl.01wm..11=wm...

OBJECTIVES, TEST MATERIALS TEST PROCEDURES and COMMENTS.

The Child.

38. (continued) Note:The division sign Acceptable Response: is not used.

Using the given numbers, the child composes, and states orally,a narrative which shows that he applies his knowledge of sharing toan environmental situation,

For example:"Father shared twelve cents amongst his four children; each received three cents."

Note: Division may also be used toanswer questions of the type "How many twos in eight?" (which is answered "four twos in eight", and recorded In the form 4x 2 W 8).

Narratives are in the form:

"If four people can go in a taxi, we will need two taxis for eight people:' "If four fish can live in one bowl, I will need two bowls for eight fish."

If desired, testsmay be devised (using sets of objects, rods, and cards with number sentences) to test this understanding, The pattern of the "sharing" tests may be followed with appropriate adaptations. Examples are given below.

a,With sets, Show the child eight objects, Divide into four twos. Say: "I have foundout how many twos in eight. Tell me a story about that,"

b. With rods. Make I, pattern with a train of red rods for the brown rod. Say: "I have foundout how many twos in eight, Tell me a story about I I that.

c.With cards. Show the child a card with the number sentence 4x 2 -1 8. Say: "This tells us how many twos in eight, Tell me a story about the NUMBER

OBJECTIVES, TEST MATERIALS TEST PROCEDURES and COMMENTS,

The Child,

AMMIIII..1ImaNOMIW.M1011101 39. Applies the know. Cards on each of which one ledge gained acid narrative is written the number facts (to be read to or by the child) learned to answer ing questions Numbers to be within the contained in given range narratives about the environment, to ten This will invoke to twenty situations embody. ing addition, sub. Note;Narratives may be traction, multipli written on the chalkboardor cation, finding presented orally, fractional parts, and division,

(a)Addition, (a) Cards, on each of which (a) Addition. is written an addition

narrative (with question), PROCEDURE:

Show the child a card withan addition narrative in question form,

For example:

"Tom has five pencils and Mary has three, Howmany pencils altogether?"

Repeat, with other addition narratives in question form,to give the required number of test items.

1 Continued NUMBER

OBJECTIVES. TEST MATERIALS TEST PROCEDURES and COMMENTS. The Child,

39, (continued) Acceptable Response:

The child should answer in one of the following forms (orallyor in writing):

"Eight." "Eight pencils." "There are eight pencils."

Note: The child may first write the number sentence usinga frame, which he then fills in, but he must then answer inone of the forms given.

(b) Subtraction (b) Cardson each of which (b) Subtraction (removal). (removal). is written a "removal"

subtraction narrative PROCEDURE: (with question),

Show the child a card with a removal subtraction narrative in question form.

For example:

"There were six birds in a tree, and four flewaway. How many birds were left in the tree?"

Repeat with other "removal" subtraction narratives in question form,to give the required number of test items,

Acceptable Response;

The child should answer in one of the following forms (orallyor in writing):

"Two" "Two birds," "There were two birds left." (Continued over) NUMBER

TEST MATERIALS TEST PROCEDURES and COMMENTS.

Note:The child may first write the numbersentence using a frame, which he then fills in, but he must thenanswer in one of the forms given.

subtraction narrative PROCEDURE: (with question).

Show the childa card with a "comparison" subtraction narrative in question form.

For example:

"Bill went to bed at six o'clock and Pamwent at eight. How many hours later did Pamgo to bed?"

Repeat with other comparison subtractionnarratives in question form to give the required number of test items.

Acceptable Response:

The child shouldanswer in one of the following forms:

"Two." "Two hours," "Pam went to bed two hours later."

Note:The child may first write the numbersentence using a frame, which he then fills in, but he must thenanswer in one of the forms given. NUMBER

TEST MATERIALS TEST PROCEDURES and COMMENTS.

1111..M

(d) Cards, on each of which (d) Multiplication. is written a multiplication narrative (with question). PROCEDURE:

Show the child a card with a multiplication narrative in question form,

For example:

"There are five houses in our street with two children in each. How many children altogether?"

Repeat, with other multiplication narratives in question form, to give the required number of test items,

Acceptable Response:

The child should answer in one of the following forms (orallyor in writing):

"Ten." "Ten children." "There are ten children altogether."

Note:The child may first write the number sentence usinga frame, which hi then fills in, but he must then answer in one of the forms given.

(e) Cards, on each of which (e) Fractions. is written a fraction narrative (with question), PROCEDURE:

Show the child a card with a fraction narrative in question form. Continued over NUMBER

OBJECTIVES. TEST MATERIALS TEST PROCEDURES and COMMENTS. The Child

39. (continued) For example:

"My rose bush had eightroses on it. I picked a quarter of them, Howman! did I pick?"

Repeat with other fraction narrativesin question form, to give the required numbe of test items.

Acceptable Response:

The child should answer inone of the following forms (orally or in writing):

"Two," "Two roses." "I picked two roses."

Note:The child may rust write the numbersentence using a frame, which hi then fills in, but he must thenanswer in one of the forms given above.

Sharing. (f) Cards,on each of which is (f)Sharing. written a "sharing" num.

tive (with question), PROCEDURE:

Show the, child a card witha "sharing" narrative in question form.

For example:

"Dad had twelve cents to share equallyamong Peter, Wendy and John. Ho* may cents did each child get?"

Continued irumOCM

OBJECTIVES, TEST MATERIALS TEST PROCEDURES and COMMENTS, The Child

39. (continued) Repeat with other "sharing" narrativesin question form, to give the required number of test items.

Acceptable Response:

The child shouldanswer in one of the following forms (orally or in writing):

"Four," "Four cents." "They each got four cents."

Note:The child may first write the numbersentence using a frame, which he then fills in, but he must thenanswer in one of the forms given above, (The number

sentence with frame, for the example above would be,12 = 3 x 1:1 or 3x =12.)

(g) Division (Howmany lots (g) If desired, the other form of division narratives may be presented. This is the form which asks, for example`How of?). many fours in eight?"

Show the child a card with a narrative in question form. For example: "Mr. Brownin making toy carts withfour wheels each, Howmany carts can he make if he has eight wheels?" NUMBER

OBJECTIVES. TEST MATERIALS TEST PROCEDURES and COMMENTS. The Child... .111Mmrir.11111.1...... 1111MIRImbr

40. Applies the (a) Materials for graph. PROCEDURE:

knowledge gained making including:

and the facts Place the materials before the child, Give the child the following information: learned to pencil and paper

representing every- graph paper (cm grid) "In our class of thirty children, fifteen children walk to school, five coo

day situations, strips of coloured by bus, five by car, four by bike and one by taxi," (a) recording the paper

in a variety of gummed squares of Say: "Choose any of the materials you like, and make a record of thi

ways and coloured paper information."

(b) intrepreting coloured pencils

the records, centicubes

counters (a) ii,Place the materials before the child. Give the child the following information:

small toys

(animals, cars etc.) "In our class of 32 children, 4 children have birthdays in January, 3wer

born in February, 2 in March, 6 in April, 0 in May, 2 in June, 0 in Jul)!

5 in August, 7 in September, 0 in October, 1in November and 2 December."

Say: "Choose any of the materials you like, and make a record of 61

information."

(a) iii. Place the materials before the child. Give the child the following information: NUMULK

OBJECTIVES. TEST MATERIALS TEST PROCEDURES and COMMENTS. The Child.. , ..111

40.(continued) "Tom threw the ball 10 metres, Wendy threw the ball 11metres, Barr) 9 metres, and John 4 metres."

Say: "Choose any of the materials you like, and makea record of dill information."

(a) iv. Place the materials before the child. Give the child the following information:

"On Monday the temperaturewas 24°, on Tuesday 28°, on Wednesday 32' on Thursday 32°, and on Friday 20°."

Say: "Choose any of the materialsyou like and make a record of dill information."

(a) v.Place the materials before the child. Give the child the following information:

"For the Stewart House collection,our class brought $9, Mr. Jones' etas: brought $4, Mr. Smith's $8, Mrs. Brown's class brought $5, Miss Green': class $7 and the kindergarten brought $10."

Say: "Choose any of the materialsyou like and make a record of fig: information."

The above, iv, constitute five test items. If tenare required, five further suitable narratives may be composed.

AcceptableResponse:

The child must indicate that he can apply his knowledge of the recording cm information in graphical form by:

(Continued over: NUMBER

OBJECTIVES. TEST MATERIALS TEST PROCEDURES and COMMENTS, The Child

40,(continued) use of a base line the selectior of a suitable unit uniform spacing of the units the correct numbers of units for each category.

Note: The recording may be made bymeans of:

linked centicubes arranged standing uptight,or lying flat lines of counters, toys, papersquares, diagrams or symbols coloured sections of graph paper.

(b) Cards (at least 30cm x (b) PROCEDURE: 30 cm in size)on each of whichone of the (b) i,Place the card before the child. following graphs is drawn and labelled as shown, Ask: "Which is the most popular pet?" "Which is the least popular?" i. Our Pets "How many pet dogs are there?" "How many more cats than mice?" Cats 3DCPD(3)3) "Can you tell me something else you have found out from this graph?" 17.14'1A7'7?-t*

Tortoisesilk

Fish cialci,011,(1

Mice 'i,

Rabbits

Birds 4 44

[Picturesdrawn or pasted on card.] NUMBER

OBJECTIVES. TEST MATERIALS TEST PROCEDURES and COMMENTS. The Child..

40, (continued)

(b) ii,Place the card before the thild, What we had for

breakfast Ask: "How many people had eggs for breakfast?" 111111 111111 "What breakfast food did most people eat?" I111111011111 "What breakfast food did fewest people eat?" 11111111X1111 "Did more peoplr, eat sausages then eggs?" 1111111111111010, I1111IWO "Can you tell something else you have found out from this graph?" 1101)101II I11111111Mill 141111WIIINII MININION11 1[01110.110110111111 KO1 Oa rd'OAn 110110111'010WI'01 MIWINDrdr0,1 0 0 0 c/ a) co 0 to a C a 8 S 150t8E to) (a co (10 02o Lu a) 0 alci.quu.2,a

Each 0 means onechild.

[Graph paper on card; objects Punters) placed on grid.]

(Continued over) NUMBER

OBJECTIVES. TEST MATERIALS TESTPP DES and COMMENTS. The Child

40.(continued)

Our Mass (b) iii. Place the card before the child.

-40kg Ask: "Who has the greatest mass?" "35 "Who has the least?" "What is the mass of the lightest child?" -'30 "Who has a mass of 32 kg ?" -25 1 "Can you tell me something else thatyou have found out from this graph: -21 IAllan 15 aIIa 1111 -II 1111lNIIIIIII -5- 1111 111aII Rill --. IIMIIIIII ID o c (i) 5 . > .7 c ca «I.,) -6 ,t1 0 0 te 4-

(Wide strips coloured in,or paper streamers pasted on card) NUMBER

OBJECTIVES. TEST MATERIALS TEST PROCEDURES and COMMENTS.

The Child ,

40. (continued) (b) iv. Place the card before the child.

Traffic Past Our Window Ask: "How many 'things' (vehicles and pedestrians) went past the window?" "How many kinds of 'things'?" AA "How many more cars than trucks?" "Were there more buses than bikes?" 0 0 "Were there more walkers and bikes than cars and taxis?" "Can you tell me something else you have found out from this graph?" 0 "Can you tell me three other things you know from this graph?" OEME 0000 000

Carsp Trucks E3

Bikes0 Taxis 0

Pedestrians U

Buses0

(Coloured paper shapes pasted on grid) (Continued over NUMBER

OBJECTIVES. TEST MATERIALS TEST PROCEDURES and COMMENTS. The Child.

40.(continued) (b) v,Place the card before the child.

Ask: "How many peoplewent on a picnic?" v. Saturday Afternoon "How manymore went to the football than the zoo?" "How many didn't leave homeon Saturday afternoon?" Went on a picnic i i "How many people watchedT.V. at the weekend?" "Can you tell me something else Went to a football match / /I you have found out about this graph?"

(It is assumed that eachperson engaged in only one of these activities.) Went to the Zoo i / i The above, iv, constitute five Worked in the garden 1/ i / 1 V test items at different levels of difficulty. Iften items are required, five furthersuitable diagramsmay be drawn and appropriate questions framed for each. The required Rayed in the park number of test items should bepresent. I i ed at a level of difficulty appropriatefor the child being tested. Watched TN, at Grandma's i Acceptable Response:

Watched TN, at home / i / The child gives the correct informationorally in response to the questions, and should be able to explain his reasoningwhen asked "How doyou know?"or Each tick / means 10 people "Why do you think that?"

If, in v. above, the childanswers incorrectly because he has not noted the fact that one tick represents ten people,his attention should be drawnto this state. melt If he then answers correctly he isat an intermediate stage. He should be retested at a later date with a similar type of test to ascertain if hecan then work without prompting.

Note: Answers to the first four questionsinv., above, are respectively 20, 20, 90, 50 (people).

1', Q NUMBER

TEST MATERIALS TEST PROCEDURES and COMMENTS. The Child

41. Uses special mathematical vocabulary.

Procedures for assessing the understanding and use of particular mathematical vocabulary, suchasfactor, product, square, prime, composite, sum difference,. .,have not.beenincluded in these test items.

Teachers should be able to discern familiarity with these terms without theuse of specific test items, SHAPE

OBJECTIVES. TEST MATERIALS TEST PROCEDURES and COMMENTS. The Child.,.

1.(a) Identifies (a) The child's classroom (a)PROCEDURE: attributes and environment, including (b) uses vocab objects which have Say: "Show me, something round," "Showme something else round," Wary to describe attributes appropriate to "Show me something straight." "Showme something else straight." lines and shapes. the child's stage of "Show me something curved,""Show me something else curved." development, "Show me something zigiag,""Show me something else zigzag. "" e.g. curved "Show me something spiral.""Show me something else spiral." straight

round Make sufficient such requests at the appropriate level, to give the requirednumbe spiral of test items. zigzag

plane Acceptable Response: solid symmetrical The child indicates the objectv, ;th the appropriate attribute.

(b) Objects in the child'sclass- (b)PROCEDURE: room environment, such as: Point to an object. child's table lid of a square box Say: "Tell me about its shape." clock face

cardboard tube (e.g. Repeat to give the required number of test items, on each occasion selectingal from lunchpaper object with a different attribute. roll)

window frame Acceptable Response: spiral cutout,e.g. snake The child describes theobject, using vocabulary of shape,e.g. curved, straight zigzag cutout mind, spiral zigzag, solid,. model of a solid shape, e.g. pyramid If the child continues torepeat the same attribute for each object (even though thi chair back is a correct response, e.g. "solid"),say: "Can you tell me something else about it knob of TN. set shape?" 1 14 (,) Orlikrt

OBJECTIVES. TEST MATERIALS TEST PROCEDURES and COMMENTS. The Child mww..,,Omi=441

2. Classifies shapes; . A set of attributes blocks. i. PROCEDURE: knows and uses

correct names for Note: In using attribute Place the blocks before the child, blocks for this test the shape square of the surface only is being Say: "Show me a square." "Show me another square." triangle considered, "Show me a triangle." "Show me another triangle." circle "Show me a circle." "Show me another circle." oblong "Show me an oblong," "Show me another oblong." "What shape is this block?" "What shape is that block?" cone

sphere The above constitute ten test items. If only five are required, the first item of each` pyramid each pair may be selected, cube

cylinder Acceptable Response prism

The child indicates a block with the correct shape, or gives the correct name for; shape, when asked.

H. The child's classroom PROCEDURE: environment. Note: The relevant Say: "Show me something in the room which has the shape of a square," "Shop tests should be given me something else which has the shape of a square." at the appropriate "Show me something which has the shape of a circle," "Show me something stage in the learning else which has the shape of a circle", (or "is circular"). process. "Show me something which has the shape of an oblong," "Show me some thing else which has the shape of an oblong," "Show me something which has the shape of a triangle." "Show me some thing else which has the shape of a triangle," "What shape is that window?" "What shape is this plate?"

(Continued over SHAPE ... OBJECTIVES. TEST MATERIALS TEST PROCEDURES and COMMENTS. The ChM.

,iYllpIIMENMI 2. (continued) The above constituteten test items. If only fiveare required, the first item of each pair may be selected,

Acceptable Response:

The child indicates objects which have the correctshape. He names correctly the shape of the objects nominated.

iii. A set of models of iii. PROCEDURE: geometric solidse.g. cones, cubes, cylinders, Place the models beforethe child. spheres, pyramids (square and triangular), Say: "Showme a sphere." "Show me another sphere." prisms (rectangular or "Show mea cube." "Show me another cube," triangular). (At least "Showme a cone." "Show me another cone," two of each.) "Showme a pyramid." "Showme a prism," Note:Models, not pictures "What shape is this?" (indicating one) of models. "What shape is this?" (indicatinganother)

It is not intended that the The above constitute ten test items, If only fiveare required, the first item of each. child use the words in pair may be selected. brackets (above),

Acceptable Response:

The child indicates objects which have the correct shape. Henames correctly the shape of the objects nominated. SHAPE 11411.01.01M. OBJEC AS. TEST MATERIALS TEST PROCEDURES and COMMENTS. The Child

2. (continued) iv.Environmental objects iv PROCEDURE: with the same shapes as the models in ii, Place the objects before the child, above, for example:

Say: "Show me something with the shape of a sphere." "Showme another." a cubeshaped box "Show me something with the shape of a cylinder." "Showme another," several dice, "Show me something with the shape ofa cube." "Show me another one." cereal packets, cigarette "Show me something with the shape ofa cone," "Show me another one." packets (rectangular "What shape has this?" "What shape has this one?" prisms) The above ,,onstitute ten test items. If only five are required, the first item of eacl basket ball, tennis ball pair may be selected. (veres) jam tins, cardboard Acceptable Response: cylinders (cylinders)

witches' hats, ice-cream The child indicates the object which shows the correct shapeor gives the correc cones (cones) name for the shape of the object indicated.

v. Cards on which is drawn v. PROCEDURE: an outline of each of the attribute block shapes, Place the cards and models before the child. one per card, (large and

small circles, squares, Say: "Some of these are plane shapes and some of theseare solid shapes." triangles, oblongs,...) ** "Point to four plane shapes." Models of geometric "Point to four solid shapes." as in iii. above; (spheres, "Is this shape plane or solid?" (indicating a plane shape.) cones, cubes, cylinders,..,) "Is this shape plane or solid?" (indicating a different shape.)

*$The test proper begins here, (continued over SHAPE

OBJECTIVES. TEST MATERIALS TEST PROCEDURES and COMMENTS. The Child .

2. (continued) The above constitute ten test items,If only five are required, the firstitem c each pair may be selected,

Note: Models, not pictures Acceptable Response: of models.

The child indicates the plane shapes andsolid shapes correctly, 11states whethe the nominated shapes are planeor solid.

vi,Cards on each of which vi. PROCEDURE: drawn a symmetrical shape or a non Place the cards before the child. symmetrical shape. e.g. Say: "Some of these shapesare symmetrical and some are not." " "Show me a symmetrical shape," "Showme another symmetrical shape' "Now another one." "Find a shape which is not symmetrical." "Find another shape which isnot symmetrical." "Now another one." "Is this one symmetricalor not symmetrical?" "How do you know?" "Is that one symmetricalor not symmetrical?" "How do you know?"

The above constitute ten test items,If only five are required,a representativ selection should be made,

Acceptable Response:

The child responds correctlyin each case. Inresponse to the question "How dov you know?" the child should indicate that he understandsthe nature of symmetr) by a response such as "Both sidesare the same." "You could fold it in half and till sides would match." "One side looks like the other side." (or similarresponse: with are notor does not for non'symmetrical shapes).

**The test proper begins here. SHAPE

TEST MATERIALS TEST PROCEDURES and COMMENTS.

i. Cards on each of which is , PROCEDURE: drawn one shape, each card being different. Place a large selection of the cards before the child,

Triangles, large, Say: "Sort these into sets of triangles, circles,squares and oblongs." small, right angled, isosceles, equilateral Ask: "Why did you put these cards in this set?"

Squares: large, small Repeat to give the required number of test items,on each occasion using adifferent selection of the cards. Oblongs: large, small, and different in proportion Acceptabll Response:

Circles: large, small. The child sorts the ,Atds correctly, and justifies the basis of his sorting. Fos example, if questioned about a small equilateral triangle and a large right-angled triangle, the child may say "They've both got three .,;Iles."

A set of models of geo- PROCEDURE: metric solids e,g, cones, Place a large selection of the models and ob is before the child. cubes, cylinders, spheres,

pyramids (square and Say: "Sort these into sets according to triangular), prisms (rectangular and triangular). Ask: "Why did you put these things here?" (At least two of each.) Repeat to give the required number of test items on each occasion using a differen. Note: Models, not pictures selection of the models. of models. Acceptable Response:

Environmental objects.with The child sorts the shapes correctly and justifies the basis of his sorting. Fo: the same shapes as the example, if questioned about a large cube and a small cube the child may sad models in 3. u (above) "They're both cubes." (Continued over SHAPE

OBJECTIVES. TEST MATERIALS TEST PROCEDURES and COMMENTS. The Child

3. (continued) for example:

a cubeshaped box several dice (cubes) cereal packets, cigarette packets (rectangular prisms) basket ball, tennis ball (spheres) jam tins, cardboard cylinders (cylinders) witches' hats, ice-cream cones (cones).

iii. The cards used in 3. i iii. PROCEDURE: above, and the models from 2, iii, of geometric Placea large selection of the material before the child. solids, e.g. cones, cubes, cylinders, spheres, Say: "Sort these into planeshapes and solid shapes." pyramids (square and triangular), prisms Ask: "Why didyou put these things in this set?" (rectangular and tri

angular). (At leasttwo Repeat to give the required number of test items, on each occasionusing a differen of each.) selection of the material.

Note: Models, not pictures Acceptable Response: of models.

The child sorts the material correctly and justifies the basis of hissorting.

Continued SHAPE

OBJECTIVES. TEST MATERIALS TEST PROCEDURES and COMMENTS. The Child...

3. (continued) iv.Cards on each of which is iv.PROCEDURE: drawn a symmetrical shape or a non- Place a selection of the cards before the child. symmetrical shape. Say: "Sort these cards into two sets, one of symmetrical shapes and one of shape: See 2 (vi). which are not symmetrical."

Ask: "Why did you put this shape in that set?"

Repeat to give the required number of test items, on each occasion usinga differen selection of the cards.

Acceptable Response:

The child sorts the cards correctly and justifies the basis of his sorting.

1" SHAPE

OBJECTIVES. TEST MATERIALS TEST PROCEDURES and COMMENTS. The Child .

4. Describes the properties of (a)-the square, (b) the triangle, (c) the circle, (d) the oblong.

(a) square (a) Cards,on each of which (a) PROCEDURE: is drawn a square.

e.g. Show the childa card on which a square is drawn.

Say: "What is thename of this shape?"

Ask: "Why is ita square?"

Repeat using other cards, to give the requirednumber of test items.

Acceptable Response:

The child gives thename correctly and describes the properties of a square ina form similar toone of the following:

"It has four equal sides and it has four right angles." "All the sides are the same length and the four anglesare the same."

If the child gives insufficientreasons, for example, "It has four sides." sayCan you tell me anything else?"

Note: The child may make statements aboutsymmetry in relation to the squaresi 1 S OBJECTIVES. TEST MATERIALS I TEST PROCEDURES and COMMENTS. The Child

(continued)

(b) triangle (b) Cards, on each of which (b)PROCEDURE: is drawn a triangle. e.g. Show the child a card on which a triangle is drawn.

Say: "What is the name of this shape?"

Ask: "Why is it a triangle?"

Repeat using other cards, to give the required number of test items.

Acceptable Response:

The child gives the name correctly and describes the properties of a triangle in a form similar to one of the following:

"It has three sides:" "It has three angles," "It has three sides and three angles."

Note: If the triangle is equilateral or isosceles, the child may make statement! about its symmetry.

(c) circle (c) Cards, on each of which (c) PROCEDURE: a circle is drawn. e.g. Show the child a card on which a circle is drawn.

Ask: "What is the name of this shape?" 0 "Why is it a circle?" (Continued over) SHAPE

OBJECTIVES. TEST MATERIALS TEST PROCEDURES and COMMENTS. The Child

4.(continued) Repeat using other cards,to give the required number of test items,

Acceptable Response:

The child gives the name correctly and describes the properties ofa circle in a form similar to one of the following:

"The outside line (circumference) is thesame distance from the centre," "If you fold it anywhere through thecentre the two sides match (aresym metrical)." "The boundary is round."

(d) oblong (d) Cards,on each of which (d)PROCEDURE: an oblong is drawn. Note: Use the word e.g. Show the childa card on which an oblong is drawn. oblonginstead of rectanglehere for Ask: "What is thename of this shape?" young children. "Why is it an oblong?"

Definitions given in the Repeat using other cardsto give the required number of test items. N.S,W. Curriculum in Mathematicsare Acceptable Response:

Rectangle: A quadri. The child gives thename correctly, and describes the properties ofan oblong in lateral with four a from similar to one of the following: right angles.

"It has four sides, the opposite sidesare equal, and all the angles are the Oblong: A rectangle same," whose adjacent "It has four rightangles and the (two)sides next to each otherare not equal." sides are unequal. Note: rl The child may makea statement about symmetry in relation to the oblong._ JJ Continued SHAPE

OBJECTIVES. TEST MATERIALS TEST PROCEDURES and COMMENTS, The Child

4. (continued)

(e) All four ) Sets of cards, (e) PROCEDURE: shapes. (a)(d).

Select five cards, from the sets (a) .(d), showinga variety of shapes, e.g. square, oblong, circle and two different triangles.

For each card say:

"What is the name of this shape?" "How do you know it is?"

If ten items are required, select ten cards from the sets (a)(d), usinga variety of shapes, and varying the proportions of the sides for the oblongs andtriangles.

Acceptable Response:

The child names the shapes and describes the properties correctly.

5. Applies knowledge The child's understanding of shapes and their properties,and his skill in applying of shape in this knowledge, comes about through constant experiments withmaterials. This environmental can be observed in the class-room through craft activities such as: situations.

paper folding (e.g. origami)

model making (e.g. windmill, witch's hat)

(Continued over) SHAPE

(OBJECTIVES. TEST MATERIALS TEST PROCEDURES and COMMENTS. The Child

5. (continued) craft tasks (e.g. covering a box with wallpaper, making a ship from scrap material)

The child's knowledge of theuse of shapes in the environment may be assessed from his answers in class discussions to questions suchas:

"Why are plate,: :ound?" "Why are wheels round?" "Why are lids of jars round?" "Why are screws round?" "Why did the Knights sit ata round table?" "Why are most lino tiles square?" "Why do the steel girders in bridges form triangles?"

"Why do rectangular wooden gates havea diagonal crossbar ?" "Why is furniture (wardrobes, sideboards) rectangular?" "Why are funnels on ships Cylindrical?"

Note: These questions are not intendedas tests of general knowledge but should follow class discussions about the practicaluse of shapes in the environment.

6. Uses special vocabulary of shape.

Procedures for assessing the understanding and use of particular geometric vocabulary such as angle, diagonal, vertical, horizontal, have no: been included in these test items. Teachers should be ableto discern familiarity with these terms without theuse of specific test items. LENGTH

OBJECTIVES. TEST MATERIALS TEST PROCEDURES and COMMENTS.

The Child.

I. Compares (a) i.Suitable moveable (a) i.PROCEDURE: (a) height objects whose height (b) length of can be measured. Place two objects before the child. two objects.

e.g. chairs Say: "Showme how to find out which one is higher (taller)." bottles Note: It may be tall boxes Repeat, using other pairs of objects, to give the required number of test items more appropriate with pencil tins Ensure that in some of the tests the difference between the objects is small younger children to On some occasions vary the instruction by saying: present tests in the

following order: "Show how to find out which one is lower (shorter)."

compares height Acceptable Response; of two objects

which can be The child places the two objects side by sideon the same surface to compare moved,(a) i. their heights, and indicates which one is higher (taller)or lower (shorter).

compares length Note: Unless the child places the objects uprighton the same surface, showini of two objects he knows the necessity for a base line, hisresponse is not acceptable. which can be

moved,(b) i. (a) ii.Suitable classroom (a) PROCEDURE: objects whose height compares height can be measured but Nominate two of the objects, of two objects which cannot be moved.

which cannot be Say: "Show me how to find out which of these is taller. You canuse anythini moved. (a) ii. e.g. bookcase you like to help you find out." bench cupboard compares length teacher's table. Repeat, nominating other pairs of objects, to give the required number of ter of two objects items. Vary the instruction on some occasions by substituting "shorter" fo; which cannot be (continued over) "taller". moved.(b) (Continued over: LENGTH

OBJECTIVES. TEST MATERIALS TEST PROCEDURES and COMMENTS. The Child

1. (continued) Measuring tools suchas Acceptable Response:

string The child compares the heights byusing one of the measuring toolsto measure tape one of the objects and transfers this measurement to theother object, by means streamer of this "intermediate measure",to enable a comparison to be made. If the child long stick does not show how to findout and merely indicates which he thinks is del woollen thread or shorter, say: "Tell me how youcan prove it."

Note, The upper part of the object must be within the reach of the child,

(b) i.Suitable moveable (b) i,PROCEDURE: objects which can be

measured Place two of the objects before the child.

e.g. books Say: "Show me how to findout which one is longer." pencils sticks Repeat, using other pairs of objects,to give the required number of test items. flat boxes Ensure that in some of the items the difference between the objectsis small. school cases. Vary the instruction by substituting "shorter" for"longer".

Acceptable Response:

The child places the two objects side by side,ends level, tocompare the lengths and indicates which one is longer or shorter. If the child does not show howto_ find out and merelyindicates which he thinks 'is longeror shorter, say: "Tell me how to prove it."

Note: Unless the child places the ends level,showing the necessity fora base_ line for comparing length, hisresponse is not acceptable. LENGTH

OBJECTIVES. TEST MATERIALS TEST PROCEDURES and COMMENTS. The Child

1, (continued) (b) ii.Suitable classroom (b) ii.PROCEDURE: objects whose length can be measured but Nominate two of the objects. which cannot be moved,

Say: "Show me how to find out which ofthese is longer. You can use anythin e.g.Chalkboard ledge you like to help you find out." child's table

window sill Repeat, nominating other pairs of objects,to give the required number of tes door opening items. Vary the instruction onsome occasions by saying "shorter", "wider" o "narrower" instead of "longer", Measuring tools such as string Acceptable Response: tape

streamer The child compares the lengths by usingone of the measuring tools to measur' long stick one of the objects and transfers this measurement to the other object, bymean woollen thread of the measuring tool, to enablea comparison to be made. If the child does no show how to find out, and merely indicates the objecthe thinks is longer shorter, wider or narrower, say: "Tellme how you can prove it."

2. Seriates by (a) Suitable moveable objects (a) PROCEDURE: comparing three whose heights can be or more objects compared Place threeor more objects before the child, (a) of different e.g.bottles heights and vases Say: "Place these in order from the tallestto the shortest." (b) of different tins lengths. Repeat, using differing numbers of objects,to give the required number of tes items. On some occasions vary the instructionby saying "shortest to tallest' "highest to lowest" or "lowest to highest" instead of "tallestto shortest".

(Continued over) LENGTH

OBJECTIVES. TEST MATERIALS TEST PROCEDURES and COMMENTS.

The Child. , nlym

2. (continued) Acceptable Response:

The child places the objects uprighton the same surface (base line) and rearrange them into the nominated order by repeatedly comparing heights of adjacen objects.

(b) Suitable moveable (b) PROCEDURE: objects whose lengths

can be compared Place three or more objects before the child,

e.g.sticks Say: "Place these in order from the longest to the shortest." pencils

rods Repeat, using differing numbers of objects, to give the required number ofte! knitting needles items. On some occasions vary the instruction by saying "shortest to longest instead of "longest to shortest".

Acceptable Response:

The child places the objectson the same base line and rearranges them into nominated order by repeatedly comparing lengths of adjacent objects. TEST MATERIALS TEST PROCEDURES and COMMENTS.

A selection of objects PROCEDURE: suitable for use as informal units, Show the child the objects with which hecan measure. e.g, pencil, piece of string, Ask: "How many handspans high is the table?"

short length of dowel "How many pencillengthsacross the window sill?" stick, a strip of "Measure the chalkboard ledge in cardboard ships." cardboard. "Measure around the wastepaper basket in stringlengths." "How far is it across the hatroom in footlengths."

Five further test items may be given if desired.

Acceptable Response:

The child carries out the measuringprocess correctly, ensuring that the successive unit; are placed so that they neither overlap nor leave spaces. He counts the units corrects}

and gives his answer orally in a form similar to the following:

"Five handspans." "Ten and a half pencillengths." "Eight and a bit cardboard strips." "Nearly two string-lengths." "Not quite eleven footlengths." LENGTH

OBJECTIVES, TEST PROCEDURES and COMMENTS. The Child, . ,

4. Estimates length A selection of objects PROCEDURE: using informal suitable for use as informal units, with units Show the child the objects with which hecan check his estimates of length. reasonable accuracy. e.g. pencil, piece of string, Ask: "How far do you think it isacross the mat, in foot-lengths?" short length of dowel stick, a strip of Say: "See if you are right." cardboard. Ask: "How high do you think the cupboard is, measured in pencillengths?"

Say: "Find out how close you are."

Say: "Estimate how long the tabletop is in dowelstick lengths." "Check your estimate."

Ask: "How many stringlengths doyou think it is around the lunch basket?"

Say: "Measure and see how wellyou estimated."

Say: "Estimate in cardboard strip lengths the height of the wall betweenthe floo and the chalkboard." "Check by measuring."

Five further test item may be given if desired.

Acceptable Response:

The child gives his estimate expressed in the nominated units.As the child checks hi; estimate, the teacher will be able to judge if this estimate is reasonablyaccurate. 1L' LENGTH

ANWIIINWmplIM.161111Wmm.....111.

OBJECTIVES, TEST MATERIALS TEST PROCEDURES and COMMENTS, The Child

5. Measures, A selection of objects suitable PROCEDURE: choosing the for use as measuring tools:

appropriate Show the childa selection of objects suitable for use as measuring tools. unit and tool, a broom handle a long piece of string Say: "Choose the best (most suitable, most sensible) of these tools to measure the a short piece of string length of the room."

ii several cardboard strips .....the length of your writing book," of differing lengths " how far it is around the waste paper tin." an orange rod the height of the T.V. set."

44 a red rod ..... the length of the light switch plate."

Five further test itemsmay be given if desired.

Acceptable Response:

The child makes his measurement correctly, selecting themost appropriate tool. Poi example, the broomhandle to measure the length of theroom, a string to measure around the wastepaper tin and the red rod to measure the light switch plate.

Note: Because the choice of tool is the important aspect of these test items, if the chilc gives a correct measurement but uses an unsuitable tool (e.g.orange rod to measure tilt length of the room) his response is not acceptable. LENGTH

TEST MATERIALS TEST PROCEDURES and. COMMENTS,

PROCEDURE:

Tell the child the followingstory. Ask the question and wait for hisresponse before proceeding to the next part of thetest.

Say: "Bill says the room isten broomsticks long, Tom says itmeasures nine pieces of string and Marysays it is forty.five pencils long. They all used their tools correctly, and yet their measurementsare all different. Why is this?"

Say: "Bill says the room is IS footsteps long, Tomsays it is 16 footsteps long and Mary says it is 17 footsteps long. Theyall measured theroom correctly with their footsteps and yet theirmeasurements are still different. Why is this?"

Ask:"How could theymeasure correctly and get the same answer?"

Ask:"Why would it be betterto measure with a metre stick?"

All the above constitute theone test item, each part of which must be answered correct. ly for success in this objective.

Acceptable Response:

The child gives answers which show that he realisesthat in the first instance the measures were made with different tookas units, and in the second, instance that not all footsteps are the same length. He states in hisown words that a uniform measure would give .a uniform answer, and that themetre is a standard measure in widespread use, LENGTH

OBJECTIVES. TEST MATERIALS TEST PROCEDURES and COMMENTS. The Child

7. Measures with a ". 20 cm rule i. Using the rule as a measuring unit, counting the number of consecutive placement standard unit. 30 cm rule to describe the length of the object measured. 50 cm rule Note: The word metre stick PROCEDURE: "rule" is used to refer tape measure to a measuring instru- Give the child an appropriate rule. Select for measurement an object longer that ment. (A "ruler" is the rule, e.g. a mat. an instrument for drawing straight Say: "Measure the length of the mat."

Repeat, nominating other objects, to give the required number of test items.

Acceptable Response:

The child carries out the activity and gives his answer in a form similar to th following:

"This table is nearly three 20 cm rules wide." "This cupboard is a bit more than two 50 cm rules high." "Our room is seven metresticks long."

Note: Itis not required that the child gives his answer in centimetres, fa example, "This table is nearly 60 centimetres wide," Such an answer is, however acceptable.

".A 20 cm, 30 cm, or Using a rule to measure an object shorter than the rule. 50 cm rule without numerals, marked in PROCEDURE: centimetre units, Give the child one of the rules. (continued over) Nominate an object shorter than the rule. (Continued over LENGTH

.M=aw...,1011111100 OBJECTIVES. TEST MATERIALS TEST PROCEDURES and COMMENTS. The Child

7. (continued) A 20 cm, 30 cm or Say: "Measure this How many centimetres long is it?" 50 cm rule with

numerals, marked in Repeat, nominating other suitable objectsand varying the rule used, to give the centimetre units, required number of test items.

Acceptable Response:

The childmeasures the object, first placing the zero mark of the rule to coincide with the "beginning" of the object, andnoting where the object "ends".

If using the unnumbered rule the childcounts the centimetre intervals.

If using the numbered rule he reads the scale.

He states hisanswer in a form similar to the following:

"The book is almost ten centimetres long."

Note: The child must correctly align thezero mark on the rule with the object before measuring. If he fails to doso his response is not acceptable.

iii. A trundle wheel. iii.Using a trundle wheel tomeasure in metres.

PROCEDURE:

Give the childa trundle wheel.

Say: "Use this to find out how long the pathis."

Repeat, to give the required number oftest items, nominating other tasks with distancesno longer than 30 metres. 14 OBJECTIVES. TEST PROCEDURES and COMMENTS. The Child A.rom.0.1=.1.111111IMINI

7. (continued) Acceptable Response:

The child uses the trundle wheel correctly,counts the clicks accurately and state: his answer in a form similar to the following:

"The path is a bit more than ten metres long."

Note: The teacher must observe the childas he carries out the action.

8. Estimates length 20 cm rule PROCEDURE: using standard 30 cm rule units. 50.cm rule Show the child the measuring tools hecan use to check his estimates. metre stick

tape measure Say: "How many metres long doyou think the room is?" Say: "See if you are right." A 20 cm, 30 cm, or 50cm

rule without numerals, Ask: "How many centimetres wide doyou think this book is?" marked in centimetre units. Say: "See if you are right."

A 20 cm, 30 cm or 50 cm Repeat, to give the required number of test items, selectingother objects, some rule with numerals, marked times asking for metre estimates, sometimes for centimetreestimates; includE in centimetre units. length, width and height.

Acceptable Response:

The child gives his estimate expressed in the nominatedunits. As the child check! his estimate, the teacher will be able to judge if thisestimate is reasonably accurate, LENGTH

TEST MATERIALS TEST PROCEDURES and COMMENTS.

1, A metre stick without i. PROCEDURE: numerals, marked in centimetre intervals. Show the child a centimetre interval on the metre stick,

Say: "How many of these on the metre stick?"

The above is the only test item to be given for this procedure.

Acceptable Response:

The child answers "one hundred ",

ii.No material is required, ii,PROCEDURE:

Ash:: "How many centimetres in a metre?"

The above is the only test item to be given for this procedure,

Acceptable Response:

The child answers "One hundred centimetres".

Note: If desired, knowledge of the fractional relationship may be tested t asking:

"What part of the metre stick is this interval?"(indicatinga centimet space)

194 "What part of a metre is a centimetre?" LENGTH

TEST MATERIALS TEST PROCEDURES and COMMENTS.

Lengths of flexible i. PROCEDURE: material

Show the child the length of ribbon fully extended, Placethe ribbon on the tab string that it assumesa curved position, for example: ribbon cord woollen thread rope,

Ask: "Is the ribbon stillas long?"or "Is there still the same length of ribbon?" "Why do you think that?"

Show the child the length of cord fully extended,Form it into a close ziga the table,

Ask: "Is the cord stillas long?"or "Is there still the same length of cord?" "Why do you think that?"

Show the child thength of string fully extended, Wind it aroundyour fingers,

Ask: "Is the string still as long?"or "Is there still the same length of string?" "Why do you think that?" (Continued LENGTH

OBJECTIVES. TEST MATERIALS TEST PROCEDURES and COMMENTS.

The Child

10. (continued) Show the child the length of wool, fully extended, Measure it on the metre rule and tell the child the measurement, for example, "fifty centimetres."

Cut the wool into two segments.

Ask: "Is there still fifty centimetres of wool?" "Why do you think that?"

Repeat any of the items, varying the material, to give the required number c test items.

Acceptable Response:

The child responds "yes" to the first question. In reply to the question "Whyd you think that?"his answer must indicate that he knows nothing has been adde or removed and only the position of the parts has been changed.

I Pairs of objects the PROCEDURE: same in length Place the two pencils before the child, parallel to each other, ends level and a spay two pencils between. two sticks D two straws two knitting needles two strips of Ask: "Are the pencils the same length?" cardboard Move one of the pencils so that the ends are not level, 196 117

Continued LENGTH

TEST MATERIALS TEST PROCEDURES and COMMENTS.

Ask: "Are the pencils still the same length?" "Why do you think that?"

Repeat, using other pairs of objects, and adapting the wording, to give the require( number of test items, on different occasions varying the amount of movement front the original position.

e.g.

0 r>

>A B

IL C

Acceptable Response;

The child responds "yes" to the first question on each occasion. If he answer "no", it is pointless to continue with the second question. LENGTH

OBJECTIVES. TEST MATERIALS TEST PROCEDURES and COMMENTS. The Child,

11. Applies knowledge The child's skill in applying knowledge of lengthin the environment can be observed of length in envir- in the classroom as the child finds answers to questions suchas: onmental

situations, "What things are sold by length?"

"What is there in the room that might beone metre wide?"

"How far is it around your waist?"

"How high can you jump?"

Further observationscan be made as the child engages in craft activities involving

measurement, and in his work with length graphs while collecting, recording andinter. preting information,

12, Uses special vocabulary of length.

Procedures for assessing the understanding anduse of special vocabulary of length such as wide, width, high, height, long, length,narrow, narrowest, rule, ruler, have not been included in these test items. Teachers should be ableto discern familiarity with these terms without the use of specific test items.

198 TEST MATERIALS TEST PROCEDURES and COMMENTS. 1'1,) A 1.

Objects of differing mass PROCEDURE: suitable for lifting, pushing or pulling by a child. Place two suitable objects before the child. e.g. rod boxes, empty Say: "Lift these to find out which one is heavier," and full attribute block boxes Place two suitable bulky objects on a smooth floor, empty and full shoebox full of pebbles Say: "Push or pull these to find out which one is heavier," shoebox full of sand empty shoebox Repeat with other suitable pairs of objects to give the required number of test items. carton of shredded pap On some occasions vary the instructions by saying carton of books empty carton to find out which one is lighter." waste paper basket ". to find out which one has the greater mass." " to find out which one has the smaller mass."

On some occasions ask the child to lift, on others to push or pull. Note: Boxes and cartons should have lids on and be The child, when lifting, may hold an object in each hand simultaneously, or may "feel" tied up with string or cord the objects one after the other. to provide a "handle" for pulling or lifting. Acceptable Response:

The child selects the correct object.

Note: The pairs of objects used in each test should bz approximately the same size, sc that the child must make his judgement based on "pull" or resistance, rather than on bulk.

199 MASS OBJECTIVES. TESTMATERIALS TEST PROCEDURES and COMMENTS. The Child...

2. Understands that Pairs of objects with the same PROCEDURE: size or shape sloes mass, but differing greatly in not determine size or shape. Place one of the pairs of objects before the child. mass. Pairs of objects the same in Say: "Lift (hold), or push or pull these to find out whichone is heavier." size or shape but differing greatly in mass. Repeat with other pairs of objects, to give the required number oftest items. If desired the substitutions of wording suggested for 1 abovemay be used.

Acceptable Response:

The child states correctly that the objects are thesame in mass, or selects the correct object if they are different in mass.

3. Understands the i. A beam balance i. PROCEDURE: use of the beam balance. Objects for placing on the Put the beam balance in front of the child. Put the plasticine ballon one of the, pans of the beam balance pans. Indicate the nails.

e.g.plasticine ball Say: "Put the nails on the other pan to obtain the samemass as the plasticine nails ball." marbles sinkers Repeat, using other materials, to give the required number oftest items, sometime rice starting with the left-hand pan and sometimes with the right-handpan. centicubes beads. 200 Continued MASS

OBJECTIVES. TEST MATERIALS TEST PROCEDURES and COMMENTS. The Child .

3. (continued) Acceptable Response: The child performs the action to obtain as nearly as possible a level beam on balance.

ii.a beam balance ii. PROCEDURE:

Objects for placing on Put the beam balance in front of the child. Place objects on the pans so that: the pans of the beam balance The right-hand pan is lower than the left, or

e.g.plasticine The left-hand pan is lower than the right, or nails marbles The pans give a level beam on the balance. sinkers rice After each placement on the pans ask: centicubes beads. "What can you tell me about the Masi of thes-i,,, things?" Repeat, using various materials in the pans, to give the required number of test items.

Acceptable Response:

The child observes the beam of the balance and responds in a form similar to the following (using the vocabulary of mass):

"The marbles are heavier than the sinkers." "The rice has less mass than the nails." "The counters and the centicubes have the same mass." (Continued over)

201 MASS

OBJECTIVES. TEST MATERIALS The Child... TEST PROCEDURES and COMMENTS. 3. (continued) Note: If the child uses size, quantityor ii .Tiber vocabulary (e.g. smaller than, less tlu ) his answer is not acceptable. For example,the following are not satisfactory. The rice is less than the sand. The marbles are smaller than the nails.

I. Measures mass A beam balance PROCEDURE: using informal units. Objects whose mass is to Put the beam balance before the child. Show himone of the objects whose mass is t be found be found. Objects to be used as Ask: "What is the mass of this object in marbles?" or informal units "How many marbles have thesame mass as this object?" e.g. plasticine Repeat, asking for the mass of other objects using differentinformal units, to give th nails required number of test items. marbles sinkers Acceptable Response: rice centicubes The child carries out the measuringprocess correctly, obtaining as nearly as possible beads level balance, and states themass of the object in the nominated units.

For example, "The block has thesame mass as ten marbles." 202 MASS

OBJECTIVES. TEST MATERIALS v ; TEST PROCEDURES and COMMENTS. The Child

5.Senates by A beam balance i. PROCEDURE: comparing the mass of two, Objects of varying mass. Place the beam balance before the Md. Give him two of the objects. three, or four objects by means Say: "Use the beam balance, and place these in order of mass from the heavie- of the beam to the lighter." balance. Repeat with other pairs of objects to give the required number of test items, some times varying the instruction by saying "lighter to heavier,"

Acceptable Response:

The child places the objects on the beam balance and then arranges them in thi nominated order.

ii.PROCEDURE:

Place the beam balance before the child. Give him three or four of the objects.

Say: "Use the beam balance to help you, and then place these in order ofmas from the heaviest to the lightest."

Make a mark inasuitable place on the child's table.

Say: "Start here."

Repeat, to give the required number of test items, using other objects, sometime varying the instruction by saying "lightest to heaviest."

Acceptable Response:

The child places the objects on the beam balance comparing two at a time, am then arranges all the objects in the nominated order. MASS

TEST MATERIALS TEST PROCEDURES and COMMENTS,

ImINII11. A beam balance, PROCEDURE:

A selection of objects whose Place before the child the beam balance andthe materirils, mass in to be eztimated, e.g.

Say: "Take this lump of plasticine inyour hands." a lump of plasticine

a large potato Give the childan empty plastic bag. a box of sand each in a small plastic bag. Ask: "How much rice do you thinkyou will have to put in this bag to give thesar mass as the plasticine?" A selection of materials to be used as informal units Say! "Use the beam balance to find out ify au are right.' e.g. nails Repeat, using other objects and otherinformal units, to give the required number beads test items. rice

marbles. Acceptable Response:

A number of empty plastic After handling the object and the units, thechild gives his estimate. Heuses the foi bags identical with those "this much rice," "this amount of nails," 'thismany counters," He is not required holding the plasticine, give a numerical answer, for example, "twelve nails." potato etc.

As the child checks his estimate, the teacherwill be able to judge if it is reasonab accurate.

204 OBJECTIVES. TEST MATERIALS TEST PROCEDURESand COMMENTS.

The Child,

Realises the need No materials needed. PROCEDURE: for standard units, Say: "Tom says this box has the same mass as 20 pebbles, Dick says it has the same mass as 25 pebbles and Harry says ISpebbles. They all used the beam balance correctly, and yet their measurements are different. Why is this ?"

Ask: "How could they measure correctly, and get the same answer?"

Ask:"Why would it be better to measure in grams or kilograms?"

All the above constitute one test item, each part of which must be answered correctly for success in this Objective, This test item is the only one required.

Acceptable Response:

The child gives answers which show that he realises that pebbles vary in mass. He states, in his own words, that a uniform unit of measure would give a uniformanswer, and that grams and kilograms are standard measures in widespread use.

3. Measures with A beam balance PROCEDURE: standard units. Centicubes Place the beam balance, the centicubes, and the mass.pieces before the child. Show him (eacii centicube has a mass one of the objects. of 1 gram) Say: "Find the mass of this object." Mass.pieces e.g. Sg, 10g, 50g, 100g, Repeat, using other objects, to give the required number of test items, 200g, 500g, 1 kg. (Continued over; (continued over)

2 0 5 MASS

minliv/M1110111Minyilr.11wAY OBJECTIVES. TEST MATERIALS TEST PROCEDURES and COMMENTS. The Child

8. (continued) Objects whosemass i3 to Acceptable Response: be measured.

The child balances the object correctlyand states his answer in standard units.As the These objects should child carries out the balancing theteacher should observe if he choosesappropriate previously be preparedso masspieces, that theycan be measured with an exact number of the available masspieces.

9. Estimates mass A beam balance PROCEDURE: using standard units, with Centicubes Place the beam balance and materials before thechild, Show him one of the objects. reasonable (each centiculle is 1gram accuracy. in mass) Ask:"What do you think is themass of this object?"

Masspieces Say. "Use the beam balance to find out ifyou are right" e.g. 5g, 10g, 50g, 100g, 204, 500g,1 kg. Repeat, selecting other objects,to give the required number of test items.

Objects whosemass is to Acceptable Response: be measured.

The child gives his estimate ingrams or kilograms. As the child checks his estimate These objects should the teacher will be Neto observe if he chooses appropriate masspieces, andto judge previously be preparedso if his estimate is reasonablyaccurate. that theycan be measured with an exact number of the available masspieces. 206 OBJECTIVES. TEST MATERIALS TEST PROCEDURES and COMMENTS. The Child

10. Conserves mass, Two identical balls of PROCEDURE: playdough, Show the child the two balls of playdough. Let him hold them, one on the palm of each hand. Ask him to tell you about the balls. Establish with him that they are as heavy as each other, i.e. that they have the same mass.

Give the child one of the balls.

Leave the other ball untouched throughout, as a reference,

Say: "Roil your ball into a `snake',"

Ask: "Has the 'snake' the same mass as the ball? Why do you think that?"

Say: "Change the 'snake' into a `pancake'."

Ask: "Has the 'pancake' the same mass as the ball? Why do you think that?"

Repeat, asking the child to reform his playdough into other shapes, to give the required number of test items, on each occasion asking if the new "shape" has the same mass as the reference ball,

Acceptable Response:

The child responds to show that he understands that his piece of playdough has the same mass as the reference ball, through all the transformations of shape.

H. Clay H. PROCEDURE: Plasticine Playdough Show the child a lump of playdough rolled into a ball.

(Continued over

207 MASS

OBJECTIVES. TEST MATERIALS TEST PROCEDURES and COMMENTS. The Child.

10. (continued) Roll the ball into a 'snake'. Ask: "Has the playdough the same mass as before?" Flatten the ball into a 'pancake'.

Ask: "Has the playdough the same mass as before? Why doyou think that?" Repeat, re-forming the playdough into other shapes, to give the required number of test items. Acceptable Response:

The child responds to show that he knows the mass remains constant throughall the transformations of shape; nothing has been added or removed. iii.Clay iii. PROCEDURE: plasticine playdough Show the child a lump of plasticine rolled intoa ball. Break or cut the ball into two pieces. Ask: "Is the mass of these two pieces the same as before?" "Why do you think that?"

Repeat, breaking the ball into three or more pieces, to give the required number of test items. Acceptable Response:

The child responds to show that he knows that the mass remainsconstant irrespec- tive of the number of pieces into which the ball is broken. 20 Continued MASS

TEST MATERIALS , TEST PROCEDURES and COMMENTS.

iv.Split peas iv, PROCEDURE: rice

beads Place before the child a plastic bag containingsome rice. Transfer the rice on to two saucers. Containers saucers Ask: "Has t 'e rice the same mass as before?" "Why do you think that?"

Return the rice to the plastic bag.

Repeat, transferring the rice on each occasion to threeor more saucers, or using other materials, to give the required number of test items.

Acceptable Response:

The child responds to show that he knows that themass remains constant irres.tc . tive of change of position.

. Two balls of clay with v, PROCEDURE: the same mass Place the ballsone on each pan of the balance to show the child the mass is the Beam balance same.

Remove the balls.

Break one ball into two pieces.

Ask: "Have these two pieces of clay still thesame mass as the ball?" "Why do you think that?"

(Continued over) MASS

OBJECTIVES. TEST MATERIALS TEST PROCEDURES and COMMENTS. The Child.

0. (continued) Re-from the clay into a ball and place beside the first ball.

Repeat, to give the required number of test items, on some occasions breaking the second ball into three or more pieces, on other occasions transforming it into shapes such as snake, sausage, pancake, spiral.

Acceptable Response:

The child responds to show that he knows that the equality of mass of the two balls has been preserved through all the physical changes of the second ball.

1. Applies knowledge The child's skill in applying knowledge of mass in the environment can be observed of mass in environ- in the classroom as he finds answers to questions such as: mental situations. What things are sold by mass?

What is there in the room that might be a kilogram in mass?

Who has the greatest mass, Tom, Dick or Harry?

Further observations can be made as the child engages in activities such as shopping play, and in collecting, recording and interpreting information when making graphs with mass measurements. 210 MASS ...gmmymmlimmiteNtmm OBJECTIVES. TEST MATERIALS TEST PROCEDURES card COMMENTS, The Child

12. Uses the special vocabulary of

MASS.

Procedures for assessing the understanding and use of special vocabulary ofmass such as:heavy, light, balance, the same mass,greater mass, smaller mass, heaviest, lightest, greatest, least, .,have not been included in these test items. Teachers shouldbe able to discern familiarity with these terms without theuse of specific test items, VOLUME (CAPACITY)

OBJECTIVES. TEST MATERIALS TEST PROCEDURES and COMMENTS. The Child

1. Understands the i. Jugs, bottles, jars,. , i. PROCEDURE: terms full and containing varying empty. amounts of coloured Show a selection of the prepared containers to the child. liquid, ranging from vote: A container empty to full. Say: "Showme the empty ones." s "full" only if it "Show me the full ones." gill not holdany store, Repeat to give the required number of test items,on each occasion varying the selection of containers.

Acceptable Response:

The child selects the appropriate containers, indicating all the empty ones, and then all the full ones. Alternatively, he may consider each container in turn and state whether it is full or empty, nearly empty, half full,

ii.Ten jugs, bottles, jars, ii.PROCEDURE:

,containing varying amounts of coloirred Show one of the containers to the child. liquid, ranging from nearly empty to nearly Say: "Make this container full." full. Repeat, using other containers to give the required number of test items, on some A bucket of water occasions saying "empty" instead of "full". a dipper and t funnel. Acceptable Response:

The child adds sufficient water to make the container full, or pours out all the watertle the container empty, whichever is appropriate, VOLUME (CAPACITY)

OBJECTIVES. TEST MATERIALS TEST PROCEDURES and COMMENTS. 44. The Child 11MP.M1 2. Measures capacity Coloured water PROCEDURE:

using informal Jugs, buckets, jars,.,. units, Tea cups, spoons egg cups, Place the coloured water, dipper and funnel before the child.

A dipper and a funnel, Ask:"How many teacupfuls ofwater will fill this jug?"

Repeat, to give the required number of test items, varying the nominatedcontainers Note: Sand or other and the informal units chosen. suitable fluid substances

may be used instead of water. Note: If the child has difficulty in carrying out the activity withwater due to spillage problems, the same activitiesmay be performed using fine sand.

Acceptable Response:

The child miles out the measuringprocess correctly, filling the container so that it will not any more, and states the capacity of the object in the nominated units. For example:

"This jugful is equal to five cupfuls."

"The capacity of this jug is five cupfuls."

"This jug will hold five cupfuls."

When the measure cannot be stated in terms of whole units the childmay state, for example:

"This jar holdsa bit more than fourteen spoonfuls." "This buckets holds nearly five jugfuls," VOLUME (CAPACITY)

OBJECTIVES. TEST MATERIALS TEST PROCEDURES and COMMENTS, The Child .=..=r....,

3. Compares the . Containers such as cups, PROCEDURE: capacity of jugs, bottles, jars, whose

containers. capacity can be compared, Show the child the coloured water. Give him two of the corl.ainers, for example, a cup and a bottle. A bucket of coloured vote:This includes: water Ask: "Which of these holds more? Which holds less? Youcan use the coloured water to help you find out." i.Direct comparison Dipper

of two containers Repeat, to give the required number of test items,on each occasion choosing two of different Funnel different containers. The containers selected should be such that the childcannot capacity. give a correct response based on visual judgement alone; for example,a very large Note: The capacity of bottle and a very small cup would not bea suitable pair for comparison. On some Direct comparison each container should occasions ask only "Which holds more?" andon others only "Which holds less?" of two containers be different from all the On others ask "Which holds more and which holds less?" Each questionprovides of the same cap- others, one test item. acity but different

shape. Acceptable Response: ii. Comparison of The child fills one container with water andpours the water into the other contain- two containers er. The water will either fill the second container and leave some over, or will all go using intermediate into the second container, leaving some space. After observing this result, the child measure (informal should be able to reach a conclusion about the relative capacity of the containers, unit). and should be able to state, for example: v. Comparison of "The bottle holds more. The cup holds less" three containers,

In response to the question "How doyou know?" the child may give reasons such as "The water would overflow if I poured it all in the cup, so the bottle holds more." "There is still a space left in the bottle after I pour ina cupful. '

214 Continued VOLUME (CAPACITY)

TEST MATERIALS TEST PROCEDURES and COMMENTS...... 11.

Note: If the child is uncertain how to proceed say, "Fill one of them with water and pour it into the other, What does that tell you?" If the child then makes the correct deduction he has reached an intermediate, stage.

1 Pairs of containers PROCEDURE: different in shape but the same in capacity. Show the child the coloured water, Give him twc containers, different in shape, but alike in capacity. A bucket of coloured

water Ask: "Which holds niore? You canuse the coloured water to help you find out."

Dipper Repeat, using other pairs of containers, to give the required number oftest items.

Funnel, Acceptable Response:

The child fills one container with water andpours it into the other, noting that the second container is now fail, and that there isno water left in the first. He then states that both containers have the same capacity, in a form such as:

"The cup and the tin hold the same amount." "They're both the same." "They hold as much as each other."

Note:If the child is uncertain how to proceed, say "Fill one of them withwates and pour it into the other. What does that tell you?" If the child then makes the correct deduction he has reached an intermediate stage, iii.Containers such as jugs, iii. PROCEDURE: bottles, jars, whose

capacity can be compared. Show the child the coloured water, dipper and funnel, also the vesselsto be used u (continued over) informal measuring tools. (Continued over) VOLUME (CAPACITY) I...11M11..VXX10

OBJECTIVES, TEST MATERIALS TEST PROCEDURES and COMMENTS, TheChild eIlk,=mmiNNIb

3, (continued) A bucket of coloured Give the child two of the containers, water, with which to rill

the containers. Ask: "Which holds more? Which holds less?"

dipper Indicate the small vessels. funnel Say: "Use one of these to measure." Small vessels, e.g.

baby tin Repeat, to give the required number of test items,on each occasion giving the child tablespoon two different containers. teacup

to be used as informal Acceptable Response: measuring tools,

The child selectsa suitable measuring tool and fills each container, counting the number of units poured into each. He then states which container holdsmore and which holds less ina form similar to one of the following:

"The jug holdsmore than the milk bottle." "The milk bottle holds less than the jug." "The jar holdsmore, the bottle holds less." "The jar holds five teacupfuls and the bottle holds four teacupfuls,so the jar holds more than the bottle."

iv.Containers such as jugs, iv.PROCEDURE: bottles, jars, whose

capacity can be Show the child the coloured water and the measuring toolsas before. compared. Give the child three of the containers, A bucket of coloured water with which to fill Ask: "Which holds most and which holds least?"

the containers. 216 Continued (continued over) VOLUME (CAPACITY)

OBJECTIVES. TEST MATERIALS TEST PROCEDURES and COMMENTS. The Child

3. (continued) dipper Repeat, to give the required number of test items,on each occasion giving the child funnel three different containers,

Small vessels, e.g. Acceptable Response: baby food tin

tablespoon The child selects a suitable measuring tool and fills each container,counting the teacup number of units poured into each. He then states which container holdsmost and to be used as informal which contains least in a form similarto one of the following: measuring tools. "The jug holds the most and thevase holds the least." "The bottle holds nine tinfuls, the jug holds ten tinfulsand the vase holds nearly six tinfuls, so the jug holds the most arid thevase holds the least."

4. senates threeor Containers such as jugs, PROCEDURE: more containers bottles, jars, whose capacity according to can be compared Show the child the coloured water, dipper and funnel. Give the childthree or more capacity. containers. A bucket of coloured

water with which to fill the Say: "Arrange these in order, starting from theone that holds the most, and ending containers. with the one that holds the least."

dipper Repeat, to give the required number of test items, varying the numbersof containers funnel given to the child, and sometimes nominating the order from leastto most.

Small vessels, e,g, Acceptable Response: baby food tin, tablespoon,

teacup, to be used as The child selects an informal measuring tool, andmeasures the capacity of each contain- informal measuring tools. er in te.s of the chosen unit. He then arranges the containers in the nominated order, basing his judgment on the numbers of units for each, VOLUME (CAPACITY) ..= OBJECTIVES, TEST MATERIALS TESTPROCEDURES and COMMENTS.

The Child.

5. Estimates capacity Containers suchas jugs, PROCEDURE: using informal bottles, jars, whose capacity

units, with reason. can be estimated Show the child the coloured water, dipper and funnel, also the vesselsto be used as able accuracy. informal measuring tools, A bucket of coloured water with which to fill Ask: "How many cupfuls do you think this bottle will hold?" the containers, Say: "Find out if you are right."

dipper Say:"Estimate haw many jugfuls this bucket will hold," fuinel "Measure and see how close you are."

Small vessels, Ask: "How many spoonfuls do you think thiscup holds?" e,g, baby.food tin. tablespoon, Say: "Check your estimate," teacup, to be used as

informal measuring tools. Say:"Estimate how many bottlefuls this jar will hold." "Measure and see if you estimated correctly."

Say: "Estimate the capacity of the milk bottle in cupfuls.' Ask: "How close was your estimate?"

The above constitute five test items. If tenare required, five further items may be presented, using variety of containers and informal units.

Acceptable Response;

The child gives his estimate expressed in the nominated units. As the childcheclu his estimate, the teacher will be able to judge if this estimate is reasonablyaccurate.

216 VOLUME (CAPACITY)

OBJECTIVES. TEST MATERIALS TEST PROCEDURES and COMMENTS.

The Child.

Understands Pairs of similar containers i, PROCEDURE: conservation with the same capacity, in relation to Fill two identical bottles with coloured water. Ensure that the child knows that capacity. Other vessels, of markedly they both hold the same amount, Leave the first bottle untouched throughout. different shapes and sizes, Pour all the water from the second bottle intoa tall "thin" container. (The volume of some with the same capacity a fluid is not as those above. Alic: "Does this (indicating the 'new' container) haveas much water in it as the changed when first bottle?" the fluid is Coloured water, "Why do you think that?" poured into different vessels.) Note: Fluid substances Return the water to its original bottle, Pour all the water from this bottle intoa such as sand may be used if *shallow "flat" container. liquids present a difficulty

because of a child's lack of Ask the same questions as before, manipulative skill.

Return the water to its original bottle, Pour all the water from this bottle intoa numberof cups.

Ask: "Do these cups have thesame amount of water as the first bottle?" "Why do you think that?"

Return the water to its original bottle.

Pour all the water from this bottle intoa variety of differently shaped small vessels.

Ask: "Do these containers have the same amount of wateras the first bottle?" "Why do you think that?"

Repeat these activities to give the required number of test items, beginning with other pairs of identical containers and pouring from the second into variously shaped containers, or into various numbers of smaller containers, (Continued over) VOLUME (CAPACITY)

TEST MATERIALS TESTPROCEDURES and COMMENTS.

Acceptable Response:

The child maintains that there is thesame amount of water in spite of change in the shape or the number of thy comainers. He indicates his understanding of conservation by stating in his own words that he knows nothing has been addedor removed, and, where applicable, that the blight of a container compensates for the width.

ii.PROCEDURE;

The procedures are as for i., but withouta "reference" container.

Fill one ,of the containers with coloured water. Pour the water intoa differently shaped container.

Ask: "Is there still as much water as before?" "Why do you think that?"

Return the water to the original container.

Repeat to give the required number of test items,on each occasion pouring the liquid into a differently shaped container,or into various numbers of smaller vessels,

Acceptable Response:

The child maintains that there is thesame amount of water in spite of change in the' shape or the number of the containers. He indicates his understanding of conservation by stating in his own words that he knows nothing has been addedor 220 removed, and, where applicable, that the height of a container compensates for the width, VOLUME (CAPACITY

TEST MATERIAL TEST PROCEDURES and COMMENTS.

PROCEDURE;

Tell the child the following story, Ask the question and wait for the response before proceeding to the next part of the test,

Say: "Ann says the bucket holds ten bottlefuls, Betty says it holds six jugfuls and Carol says it holds twenty tinfuls, They all measured correctly and yet the numbers are different, Why is this?"

Say: "Dot says the jug holds fifteen spoonfuls, Emma says it holds twenty-one spoonfuls and Fred says it holds sixty spoonfuls, They each brought their own spoon from home and they all measured correctly and yet the numbers are all different. Why is this?"

p Ask:"How could they measure correctly and get the same answer?"

Ask:"Why would it be better to use a litre measure?"

All the k ;:.; constitute one test item, each part of which must be answered correctly for success in this Objective. This test item is the only one required,

Acceptable Response:

The child gives answers which show that he realises that in the first instance the measures were made with different measuring units, and in the second instance thatnot

all spoons hold the same amount, He states in his own words that a uniformmeasure would give a uniform answer, and that the litre is a standard measure in widespread use, VOLUME (CAPACITY)

OBJECTIVES. TEST MATERIALS TEST PROCEDURESand COMMENTS. The Child

8, Measures with A litre jug. PROCEDURE: a standard unit

eitre). A number of containers Show the child the coloured water, funnel, dipper and litre jug. with capacity of a litre or more. Nominate one of the containers,

A bucket of coloured water, Say: "How many litres does this jar hold?" funnel, dipper. Repeat, to give the required number of test items, nominating other large containers whose capacity is to be measured in litres.

Acceptable Response:

The child finds the capacity of the container, by correctly filling and refillingthe litre measure, and counting the number of litres poured into the container. He states the capacity to the nearest litre, in a form similar toone of the following:

"The jar holds nearly two litres."

"The bucket will hold a little over four litres."

"The vase holds about one and a half litres."

"The capacity of the bottle is one litre,"

222 vi../LUmt (CArAt.a 11'1

OBJECTIVES. TEST MATERIALS TEST PROCEDURES and COMMENTS. The Child ...

9. Estimates capacity A litre jug PROCEDURE: using a standard unit (litre), with A bucket of coloured water, Show the child the coloured water, dipper, funnel, and litre jug. reasonable dipper, funnel. accuracy. Ask: "How many litres do you think this holds?" ("will hold?") Containers whose capacity is to be estimated. Say: "Measure and find out if you are

Repeat, nominating other containers, to give the required number of test items.

Acceptable Response:

The child gives his estimate expressed in litres. As the child checks his estimate, the teacher will be able to judge if this estimate is reasonably accurate.

10. Applies know- The child's skill in applying knowledge of capacity in the enviconrnent can be observed ledge of capacity as the child: in environmental situations. carries out classroom tasks, e.g. fills flower vases, fills painting jars

engages in activities which involve using or measuring fluid and liquid substances, such as

afternoon-tea party dramatisation playing shops using recipes for cooking simple foods.

discusses articles sold by capacity (Continued over)

223 VOLUME (CAPACITY

OBJECTIVES. TEST MATERIALS TEST PROCEDURES and COMMENTS,

The Child,

10, (continued) xusses commodities sold by fluid measures

makes or interprets graphs based on capacity

finds out how much liquid (or fluid) is ina partly filled container (by measuring with an informal unit suchas a cup, or by measuring with a formal unit such as a litre jug, or by pouring intoa calibrated vessel and reading the scale).

Answers may be given in the from:

"about three and a half cupfuls" "nearly a litre" "a little over half a litre" "the scale says '600 ".

11. Uses special vocabulary of capacity.

Procedures for assessing the understanding anduse of special vocabulary of capacity, such as holds more, holds less, holds least, capacity,

.110, liquid, have not been included in these test items. Teachers should be ableto discern familiarity with these terms withoutthe use of specific tests

The word volume is not used at this stage. The concept of capacity only is dealt with. (Hencetests to find the volume of liquids in partly filled vessels are not included. In class acti%jties childrenare asked "How much water is in the bucket?" but not "What is the volume of water in the bucket?") 294 VOLUME (CAPACITY)

OBJECTIVES. TEST MATERIALS TEST PROCEDURES and COMMENT!).

The Child.

Note: Capacity refers to the amount of substancea vessel holds when full (or will hold, when full).

Volume refers to the amount of substance of the container itself plus theamount of substance it holds (or will hold) when full.

Thus the capacity of a bottle is the amount of substance (air, fluid, etc.)it is capable of holding; the volume of the bottle is the amount of glass of which itis made plus the amount of substance it is capable of holding.

(Briefly: capacity is contained volume.)

Capacity is usually expressed in liquidmeasures, e,g, litre, millilitre.

Volume is usually expressed in cubicmeasures, e.g. cubic metre, cubic centimetre,

225 AREA

OBJECTIVES, TEST MATERIALS TEST PROCEDURES and COMMENTS. The Child,

I, Understands the Classroom objects with flat 1, PROCEDURE: term "crrt Ping surfaces:

a surface Have the table top clear ofany objects, e.g. Table

Note: "Covering" Bookshelf Say: "Showme the surface of this table." in this mathematical Mat sensemeans having Picturebook Repeat for the shelf, mat, picture book, . ,to give the requirml er ni no overlap of units test items, and leaving no space between units. Acceptable Response:

The child must run his hand over the entiretop surface of the object. If he mer$ points, or places a finger on the surface inone place, his response is not acceptable. Probe further by asking "Is that all of the surface?"If the child then indicates t surface correctly, hisresponse may be accepted.

PROCEDURE:

(a) A classroom mat (a) Use the classroom mat and the sheets of artpaper.

Sheets of artpaper (a few Place the maton the floor. Give the child the sheets of paper. more than are necessary to cover the mat) Say: "Cover the surface of the mat with this paper."

(b) A picture book ofquarto Repeat the procedure, adapting the words of theinstruction, using in turn size or larger

(b)the picture book and playing cards Playing cards (a fewmore (c)book shelf and exercise books than necessary tocover (d)table top and square lino tiles the book) table top and round icecream container lids. 29 re)) continued AREA

OBJECTIVES. TEST MATERIALS TEST PROCEDURES and COMMENTS.

The Child,

1. (continued) (c) A book shelf (clear If ten test items are required, five further similar items may be presented, of objects)

Acceptable Response: Uniform exercise books (a few more (a) The child places the sheets of paper on the surface of the mat (with reasonable than necessary to cover accuracy) so that the sheets do not overlap each other andno spaces are the shelf) left between sheets, It may be necessary, for complete coverage of the surface, foi parts of sheets to extend beyond the boundary. (d)A table (top clear of objects) (b), (c), (d)Similar responses are required, using the different materials,

Squire lino tiles (a few (e) The child places the lids as closely as possible on the table top, finds that he car more than are necessary make the lids touch without overlapping, but that always spaces are left. to cover the tabletop) He then must state that the surface is not covered, He may indicate a further lever (e) A table (top clear of of understanding by a response such as "You can't cover it with that shape." objects) If he places the lids over the table top but makes no statement, prompt him b) Lids of round asking: "Have you covered the surface?" icecream containers

(sufficient in Note: A child's response is not acceptable if there is overlapping of units (see definitior number to carry ofcovering,page 212). If the tester is uncertain whether the overlapping of units tie; out the test) place because

the child feels it is wrong to place parts of units outside the boundary, or the units on the boundary "fall off' unless the child overlaps them (in b, c, d or e)

then these test items should be repeated, using the same units to cover a surface o: similar dimensions at floor level, AREA

OBJECTIVES. TEST MATERIALS TEST PROCEDURESandCOMMENTS. The Child

2. Covers a surface Surfaces to be covered with PROCEDURE: using informal informal units units, and dis- Place the prepared art paper before the child. Give him the matchboxes. covers the num- Objects to serve as informal berof units units. Say: "Cover the surface of the paper with matchboxes and find out howmany are required (i.e. needed to do this." finds the area). Note: These materials must be prepared in advance, so Repeat, using other prepared materials, to give the required number of test items. that an exact number of units cover the surfaces to be Acceptable Response: measured. Several extra units of each type are to be The child covers the surface correctly with the nominated units, and counts the number included in the set given to used. the child. (He may also, if asked, make a statement about thearea of the surface being measured. Examples of suitable surfaces For example: "The area of the paper equals fifteen matchboxes.") and units are: Note: Estimation activities may be given, based on these tests, if desired. A sheet of art paper cut to an appropriate size The same types of materials are required. The procedures given above would be varied Matchboxes as units so that the tester says, for example: (see note above) "How many matchboxes do you think would be required to cover the surface of A sheet of cardboard cut this paper? Check with the matchboxes to see if you are right." to an appropriate size Uniform exercise books For an acceptable response, the child gives his estimate expressed in the nominated as units (see note above) units. As the child checks his estimate, the teacher will b able to judge if this estimate is reasonably accurate. A piece of felt cut to an appropriate size Playing cards as units 228 (see note above). AREA

OBJECTIVES. TEST MATERIALS TEST PROCEDURES and COMMENTS.

The Child

3.Chooses a unit PROCEDURE: appropriate to the Small heaps of materials

area of the gurr suitable for measuring units Show the child the heaps of different materials suitable for measuring units,

face to be (one type of material only

measured. in each heap) Ask:"Which one of these could you use as units to find the area of

the table-top?" For example: this picture book?" the hatroom?" linotiles 30cm square that wall chart?" a small envelope?" small uniform ceramic tiles (about 3 cm square) On each occasion, ask:

playing cards "Why did you choose that one?"

sheets of art paper The above constitute ten test items.

Acceptable Response:

The child names an appropriate unit. For example:

linotiles or sheets of art paper for the hatroom ceramic tiles for the envelope or book playing cards for the book or wall chart.

In re3ponse to the question "Why did you choose that one?" he gives a response similar to one of the foil wing:

"You Q.)n't 4d as many :hose _he at= a." "You could cover the surface more 4:kly with On" "It would take too long to put playing cards on the hatroom floor," "You could t NO, the sheets of paper easily," "The envekin ! so small you a small *it?

229 AREA

OBJECTIVES. TEST MATERIALS TEST PROCEDURES andCOMMENTS. The Child

4. Finds shapes that Ten (or more) identical "tiles" PROCEDURE: tesselate to make of each of the following a surface. shapes: Show the child the sets of "tiles",ten (or more) to a set,

square Say: "Take each set of 'tiles' inturn and fit them together to find out which shapes Vote: Theuse of the oblong will make a surface (i.e, tesselate)," Word tesselateis not parallelogram iecessary for the tests; irregular quadrilateral Repeat for each shape inturn, to give the required number of test items, Ensure that he words "fit triangle some of the shapes chosen do not tesselate, ogether",or "fit pentagon ogether to make a hexagon Acceptable Response: urface"may be used. octagon circle The child carries out theinstruction with each of the shapes inturn, turning the 'tiles" semi-circle upside down wherenecessary to make them "fit", He then states that thesquares, oblongs, parallelograms, quadrilaterals, trianglesand hexagons will tesselate, and that Note: "Tiles" may be made the pentagons, octagons,circles and semi-circles will not tesselate. Hemay use his own of plastic, wood, strongcard- words instead of the word "tesselate", board, stiff paper, foam rubber or other suitable materii. Note: If the child places all the "tiles" froma set in one, or two, long rows, say: The size should be suchthat the ten can be arranged on "Can you do ita different way?"or the child's table. "Can you makea different shape?"

in order to see if he can make a more compact surface, This will allowa better assess. ment of his abilit; to tesselate,

230 AREA .....MIIRP.RPMe=111

TEST MATERIALS TEST PROCEDURES and COMMENTS.

Thirty.two identical square i. PROCEDURE: "tiles" made into two shapes: Show the child the two shapes made from the "tiles".

a squaremade of 16 "tiles" (four each way) Ask: "Which of these has the greater area?"

Ail oblong made of two "Why do you think that?" rows of eight "tiles" Break up the arrangements and remake into twoother shapes, each containing sixteen "tiles".

Repeat, to give the required number of test items, on every occasion using different pairs of arrangements with sixteen "tiles" each.

Acceptable Response:

The child states on each occasion that both shapes have the same area, becausethey both contain the same number of squares.

u.PROCEDURE:

Thirty.two identical square Show the child the two squares each made of sixteen "tiles". Put one square in "tiles", made into two front of the child, one in front of the tester. square shapes, each ofsixteen "tiles". Say: "This square is mine. That one is yours. They both have the same area.Make your square into another shape."

"Which has the greater area now, your shape or mine?" "Why do you think that?"

(Continued over)

231 AREA ...lam.m...... mn...... OBJECTIVES. TEST MATERIALS TEST PROCEDURES andCOMMENTS. The Child

5. (continued) "Make yours into another shape. Now which has the greaterarea, yours or mine?" "Why doyou think that?"

"I will make ; 'y square into another shape, Now whose shape hasthe geater area?" "Why?"

Note: The rearranged shapesmay be quite irregular in outline.or example:

All the above constitute one test item, There must be correctresponses to each part for success in tlis Procedure. Onlyone test item is required.

Acceptable Response:

The child indicates hisunderstanding of conservation by stating that each shapehas an area of sixteen squares, and the two surfaces remain equal inarea, no matter= how theyare rearranged.

iii, PROCEDURE:

Two squares of centimetre Show the child thetwo cards. graph paper each 4cm by 4 cm, colouredgreen, and Say: "These are two paddocks belonging to two different farmers.They aregreen mounted on white card, becatse they are covered in green grass, Each farmer put four shedson his pasture (paddock). The first farmer built continued them like this," (Place fourcubes 232 Continued AREA

OBJECTIVES. TEST MATERIALS TEST PROCEDURES and COMMENTS,

The Child.

aOI

(continued) on the grid, randomly scattered, none touching.) "The second farmer built his like this," (Place four cubes on the grid so that they touch,

or 1:10 or or IR &PIA

Eight cubes, each 1 cm x 1 an Ask: "Which farmer has more grass left?" or x 1 cm "Which farmer's cows will have more grass to eat?" (White rods are suitable. "Why do you think that?" Centicubes may be used.) Re-arrange the four "buildings" on each of the two "paddocks" to give diffet placements. Repeat the questions,

Continue in this way, to give the required number of test items,

AcceptableResponse:

The child indicates his understanding of conservation by stating that the grassed parts of both paddocks are the same in area no matter now the buildings are arranged or re-arranged, because on each occasion the buildings occupy four squares, and since the paddocks were originally equal in area, the area of the grassed parts will be equal. te:Understanding A number of identical shapes iv,PROCEDURE: :onsmation with- (squares, oblongs or triangles) : use of informal cut out of stiff paper. Take two of the paper shapes; ensure that the child understands that they are equal ts may also be in area by superimposing one on the other. Put one shape aside as a "reference" ted. shape. Cut the other into two pieces using a straight, curved or zirag cut. Arrange the two pieces apart, or to form a "new" shape,

(Continued over)

233 AREA

TEST MATERIALS TEST PROCEDURES and COMMENTS.

Ask: "Is thearea now the same as that of this first shape?" (Indiate the "reference" shape.)"Why doyou think that?"

Repeat, to give the required number of test items, using the othei identicalpaper. shapes, cutting into two pieces withdifferent cuts,or into three pieces or four I pieces. Repeat the questions on each occasion, to compare thearea with the "reference" shape.

Acceptable Risponse:

The child indicates his understanding of conservation bystating that thearea remains the same no matter how the surface is rearranged, since nothingis added or removed.

Paper shapes v.PROCEDURE: (circle,square,

oblong, triangle). A similar activity is carried out, without using a "reference" shape. Takeone of the shapes. Show it to the chili Cutinto two pieces, andrearrange.

Ask: "Is thearea the same as before?" "Why doyou think that?"

Repeat with other shapes and differentcuts to give the required number oftest items.

Acceptable Response:

As above. The childmay "prove" the area is unchanged by reforming the original shape.

234 TIME

TEST MATERIALS TEST PROCEDURES and COMMENTS,

PROCEDURE:

Ask: "Is this day time or night time?" "Is it morning or afternoon now?"

Say: "Tell me something we do in the morning." "Tell me something we do in the afternoon "

"Tell me something you do at night."

The above constitute five test items. If ten are required the questions may be repeated at different times of day.

Acceptable Response:

The child responds correctly to each question, and names activities that relate tothe nominated times in such a way that the tester knows that the child understands the time

divisions.

If the child gives a roponse such as "play", "eat" or "skip", which could apply to any of the time divisions, the tester may be une,ettain whether the child is referring to the

specified time. In such a case say:

"Tell me something that we do only in the morning (afternoon, night)."or

"Tell me something else we do in the morning (afternoon, night)."

235 TIME

OBJECTIVES. TEST MATERIALS TEST PROCEDURES andCOMMENTS. The Child.

2. Tells the time i. PROCEDURE: from "traditional" A "traditional" clockor and digital clocks, clock.face Show the child the traditionalclock, Set the hands ata selected time, and relates times

to daily events. A digital clock Ask: "What is the timeon the clockface?' (twelve hour)

Repeat, to give the required number of test items at the desiredlevel of difficulty, setting the hands at differenttimes on each occasion.

Note: Suggested Acceptable Response: order of presentation:

The child states thecorrect time, a, o'clock b. half past PROCEDURE: c. quarter past d, quarter to Show the child the digitalclock, or draw e. five minute a digital clockface, showinga selected time. intervals past e.g. or to the hour, =144.M.ImI.plYYMi. 10 55

Say: "Read the timeon the clock."

** Ask: "What does this mean?"

If the time shown is past the half hour (for example 10.55), whenthe child has read 1 the numerals,say:

"Can you tellme this another way?"

236

*1The test proper begins here. Continued TIME

TEST MATERIALS TEST PROCEDURES and COMMENTS.

"1.114P.M.1.1111=11.10.100.1.111111MINO

Repeat, to give the required number of test items at the desired level of difficulty, showing different readings on each occasion.

Acceptable Response:

When the child reads the figures shown, for example, "ten fiftyfive ", he states that this means "fifty.five minutes past ten", and that the time can also be stated as "five (minutes) to eleven".

ill PROCEDURE:

Repeat the procedure for i and/or ii, and when the child has stated the time given on the clock:

Ask: "If this were daytime (night time), what might you be doing?"

Repeat, to give the required number of test items, using other times within the selected level of difficulty.

Acceptable Response:

The child names activities appropriate for the time stated. TIME

OBJECTIVES. TEST MATERIALS TEST PROCEDURES and COMMENTS, The Child

.===mmIn11Nme.mmoMmENINo 3, (a) Knows the (a) PROCEDURE; days of the week, insequence, Ask: "Can you tellme the days of the week in order?" (b)knows the months of the "Can you do it startingat Wednesday?" yeu In sequence, "Can you do it startingat Saturday?" (c) usesa calendar, ind "What day comes after Thursday?" (d) mites dates, "What daycomes before Monday?"

If further test itemsare required, ask other &imilar questions, to give the required number.

Acceptable Response;

The child answers each question correctly.

(b) PROCEDURE;

Say: "Tell me the months of theyear in order,"

Ask: "Whatcomes after May?" "What comes after December?" "What comes before July?" "What comes before January?"

If further test itemsare required, ask other similar questions, to give the required number of items.

238 Continued TIME

OBJECTIVES. TEST MATERIALS TEST PROCEDURES and COMMENTS. Tho Child, . ,

3. (continued) Acceptable Response:

The child answers the question correctly.

Pencil and paper Give the child the calendar,

Ask: "What year is that calendar for?"

Say: "Show me to-day on the calendar." "Show me the 10th June. What day of the week is it?"

Ask: "What is the date of the first Tuesday in March?"

Say: "Say the dates of all the Sundays in January." "Find out how many days thereare in November,"

Ask: "What is the sixth month?"

Write on the chalkboard, "4th December,*1977". Say: "Show me this day on the calendar."

Write on the chalkboard "11.5.77" (or "11.5.*1977"). Say: "Show me this day on the calendar."

Ask: "What day of the week is Christmas Day this year?"

The above constitute ten test items. If only five are required,a representative selection should be made.

' Write numerals for the tunent year instead of 1977. (Continued over) TIME

OBJECTIVES. TEST MATERIALS TEST PROCEDURES and COMMENTS. The Child

3. (continued) Acceptable Response:

The childanswers the question correctly.

(d) A calendar (d) PROCEDURE:

Pencil andpaper Show the child the calendar, Give him thepencil andpaper,

Say: "Write today's date." "Write the date of next Monday." "Write the date of all the Saturdays in June," "If the next Infants' School Club meetingis the fourth Thursday of this month, write what the date will be." "Write this date anotherway: 6.7.1977" "Write this date anotherway: 2nd Februaty,*1977,"

If further test itemsare required, ask other similar questions, to give the required number of items.

Acceptable Response:

The child writes the date correctly,

210

$ Write numerals for the currentyear instead of 1977. TIME

OBJECTIVES. TEST MATERIALS TEST PROCEDURESand COMMENTS. The Child ,

4. Understands PROCEDURE: dapped time, and

uses related vocab (a) Say: "If John runs around the playground and Bill walks, and theyboth star &TY, together, who will finish first? Why?"

Note: Related (b) Two pictures, one of (b) Recall an old anda new building in the environment, or show the child the tw vocabulary: an old building and pictures, one of an old building and one of a new building. (a) fast /slow one of a new building,

old /new the age difference being Say: "Which building is old and whichone is new?" (c) old/young quite obvious. "Why is that one older?" (d) eallYilate

(e) before/after (c) A picture of a child, and (c) Show the child the two pictures,one of a child and one of an adult. a picture of an adult.

Say: "Which person is old and which one is young?" "Why is that one younger?"

(d) Say: "The bus was due to arrive at nine o'clock, and itcame at ten past nine quarter to ten, five to nine, nine fifteen, eight forty.five). Was the bus earl or late? Why do you think that?"

(e) A set of pictures (about (e) Spread out in front of the child theset of pictures, in a random arrangement. five) each of which shows a stage in a sequence, Say: "Put these in the right order," e.g. getting ready for "Why did you put thisone first and this one next? Why did you put tit school. one next? Why did you put this one last?"

The above constitute five test items. If ten are required five similar itemsmay b devised, ofte each of the typesa, b, c, d and e.

(Continued oven

241 TIME

OBJECTIVES. TEST MATERIALS TEST PROCEDURES and COMMENTS. The Child OPMww., .M...... 1010111MMIN 4, (continued) Acceptable Response;

The chili answers the questions using appropriate words. Inresponse to the questiol "Why ?", the child gives answers similarto the following.

(a)"Running is fast, walking is sloe "Running is faster than walking." "Walking is slow, running is fast." "Walking is dower than running."

(b) "Itwas built fast" "It has been up a longer time, so it's older."

(d) "The time it came is before thetime set down, so it is early.""It came after i was supposed to, so it was late." "Ten past nine is later than nine o'clock." "Eigh forty.five is earlier than nine o'clock."

(e) "This happens before that." "Youput on your socks before your shoes." "Waving goodbye is the last thing." TIME

TEST MATERIALS TEST PROCEDURES and COMMENTS.

*11.11=1

PROCEDURi A clock or clock-face (a) Say: "When I left home to come to school the clock showed eight o'clock. When I A calendar, got to school it showed nine o'clock, How long did it take me?"

(b) Say: "I started writinga letter at eleven o'clock, It took me half an hour,Show me on the clock the time when I finished my letter."

(c) Say: "Mum put the biscuits in the oven at ten forty-five. They took a quarter of an hour to cook. What time did she take them out?"

(d) Say: "Thiswas the time when Mary began her painting.

This was the time when she fmished,"

(Set the handson the clock-face to show a quarter to twelve, then move them to show ten to twelve.)

"How many minutes did the painting take?"

(e) Say: "On Saturday Mum goes to work atseven thirty a.m. She comes home at noon, How long is she away from home?"

(0 Say: "Today,is 15th September and Mary's birthday ison this day (point to 20th September on the calendar): How many days to Mary's birthday?"

(Continued over) TIME

OBJECTIVES, TEST MATERIALS TESTPROCEDURES and COMMENTS. The Child

5. (continued) (g) Say: "How many weeks till the holidays?"(Ensure that the child knows the present date and the starting date of the holidays,and can find these on the calendar,)

(h) Say: "Mummy is going fora holiday with Grandma. She will be away from the beginning of June to the end of July. Howmany months is that?"

(i) Say: "We breakup on the 26th August, and come back to schoolon the 12th September. Showme that on the calendar. How many days arewe away from school?"

(j) Say: "John is goingto Melbourne on the 17th October. He will beaway fifteen days. What will be the date of his return?"

The above constitute ten testitems. If only five are requireduse b, d, f, h and

Acceptable Response:

Using the clock or the calendar, the childanswers correctly.

6 Uses special vocab ulaty of time.

Procedures for assessing the understanding and use of special vocabulary of time, suchas second, minute, hour, calendar, date, week, month, year, midday, midnight, noon, afternoon, morning, evening, day, night, , have not been included in these test items.Teachers should be able to discern familiarity with theseterms without the use of specific tests. 2114A MONEY

OBJECTIVES. TEST MATERIALS TEST' PROCEDURES and COMMENTS.' The Child,. .=MIME.

1, Recognises A collection of different i. PROCEDURE; coins. Australian coins in current use, lc, 2c, 5c, 10c, 20c, 50c, Show the child one of the coins.

Ask: "What is this coin?"

Repeat, using each of the other five coins in random order, showing four of them twice, to give ten test items.

Acceptable Response:

The child names each coin correctly.

ii.PROCEDURE:

Place before the child the collection of coins, spread out randomly.

Say: "Show me a two-cent coin,"

Repeat, naming other coins, to give the required number of test items,

Acceptable Response:

The child indicates the correct coin, MONEY

TEST MATERIALS TEST PROCEDURES and COMMENTS. ."61.....11 A collection of Australian PROCEDURE: coins of different denomina- tions. a.Place a large selection of the coins before the child,

Ask: "Which coin is worth the most?" "Which coin is worth the least?"

b.Ask: "How many one-cent coins equal this?"

(Indicatea two-cent coin, five-cent, ten-cent, ,.)

"How many two-cent coins equal this?" (Indicate the ten-cent coin, then the twenty -cent.)

c. Show the child two different coins.

Ask: "Could I buy more (less) with this,or with this?"

d.Show the childa combination of coins, e.g. a ten-cent and a two -cent, or a five- cent, a two-cent and a one cent coin,

Say: "Showme this amount another way."

e.Give the child all the coins.

Ask: "Canyou show me some different ways of making up twenty cents? Te cents? Fifty cents? Five cents?"

The above constitute five test items. If tenare required, each type, a, b, c, d, and may be repeated using other coins.

Acceptable Response: 24 The child indicates by his response that he understands the value of the coins. IVIVINtY

OBJECTIVES. TEST MATERIALS TEST PROCEDURES and COMMENTS. The Child .

Counts coins to A collection of various coins PROCEDURE: find the amount in current use. of money. Give the child a selection of loins whose total amount is within the range of (a) ten cents, (b) twenty cents, (c) fifty cents, (d) ninety-nine cents.

Say: "Count these and tell me how much money there is."

Repeat, using other selections of coins within the same range, to give the require number of test items.

Acceptable Response:

The child counts the coins, and states the total amount correctly. Where appropriate the child may first sort the coins.

Uses appropriate A collection of coins*with PROCEDURE: coins for pu- which to make purchases. chasing. Say: "This ------costs twelve cents. Show me the coins you would use to buy it.' Objects with price tags. Show the child a price tag (within the selected range). Amounts will be within the range of: Say: "Show me the coins you would use to buy this article."

a. ten cents Repeat, using instructions of both types, to give the required number of test items fa b. twenty cents the selected range. c. fifty cents d. ninety-nine cents. (Continued over

247 MONEY

OBJECTIVES. TEST MATERIALS TEST PROCEDURES and COMMENTS. The Child

4. (continued) Acceptable Response:

The child selects coins to makeup the designated amount. If the child selectsas inappropriate number of smaller coins to makeup the amount, say:

"Can you makeup the amount using fewer coins?"

Note: For prices within therange of ten cents, it is less important that the chilc selects the fewest coins.

5, (a) Understands A collection of coins (a) PROCEDURE: the concept of with which to carry out change and transactions. Say: "You be the shop-keeper. Iwant to buy this lolly for fifteen cents. Here i (b) carries out my twenty cents." (Offer a twenty-cent coin.) shopping trans. Objects and price tags actions. Repeat, with suitable variations, to give therequired number of test items, withit Amounts will be within the the selected pricerange. range of:

Acceptable Response: a. ten cents b. twenty cents The child gives the lolly andan amount of change. Provided that, change is given. c. fifty cents the response is acceptableeven when the amount of change is incorrect. d. ninety-nine cents e, one dollar (after Note:The child may give the change by usinga subtraction method, or the shop Objective 6.) keepers' method of "counting on." 248 TEST MATERIALS TEST PROCEDURES and COMMENTS.

, .

(b) PROCEDURE:

Say: "You be the shopkeeper. Iwant to buy this------for eight cents Here is my ten cents."

Say: "You be the shop.keeper. Iwant to buy this------for seven cents Here is my five cents."

Say: "You be the shopkeeper. Iwant to buy this----- for fifteen cents Here ismy money." (Give the child the correct amount)

Repeat using other similar shoppingsituations, involving from the child

no change, or change, or a request for more money,

to give the required number of test items, withinthe selectedrange.

Acceptable Response:

The child carriesout the transaction correctly,

giving the correct amount of change,or

requesting the required additionalamount,or

recognising that the exact amount has beentendered.

(Continued over)

249 MONEY

OBJECTIVES. TEST MATERIALS TEST PROCEDURESand COMMENTS.

The Child.

MIL

5. (continued) Note:Change may be given using a subtraction method, or by "counting on", as shopkeepers do. The shopkeepers' method isquicker and more efficient, but the child will not know the actual amount of change given.

H. A selection of objects H. PROCEDURE: with the following price tags: Show the child all the objects with price tags. Give him the coins.

a. 8c, 7c, 9c, Sc Say:"I'll be the shop-keeper and you be the buyer. Which of these would you b. 15c, 12c, 17c, 20c like to buy? Give me the money to buy it." c. 32c, 45c, 21c, 16c d. 98c, 33c, 76c, 84c Repeat, to give the required number of test items within the selected range.

A number of five-cent, Acceptable Response: tencent, twenty-cent and fifty-cent coins. The child tenders an amount of money greater than or equal to the price. When change is given (by the tester) the child checks the amount to see that it is correct.

Note:When a larger amount is tendered it should be the closest possible that can be made with the available coins. Thus, if the cost of the article is forty-six cents, ytt) ul7r1; then fifty cents should be tendered rather than, for example, ninety cents.

6. Recognises the A one-dollar note PROCEDURE: dollar note and the twoadollar, A two-dollar note Show the child one of the notes. note. Ask: "What is this note?" 25U Shrs the child the other note. Ask: "What is this note?" MONEY

OBJECTIVES. TEST MATERIALS TEST PROCEDURES and COMMENTS. The Child ...

6. (continued) Place the two notes before the child. Say: "Show me the dollar note." "Show me the two-dollar note."

The above are the only test items for this objective.

Acceptable Response:

The child names each of the notes correctly. He indicates the correct noteon both occasions.

7. Reads and uses Price tags with amounts PROCEDURE: symbols for within the following ranges: recording Show the child, one at a time, the price tags within the selectedrange. amounts. a. 8c, 7c, 9c, 5c b.15c, 12c, 17c, 20c Say: "Read this." c. 32c, 45c, 21c 16c d.98c, 33c, 76c, 84c Give the child pencil and paper. Name an amour t within the selectedrange, for e.over $1 and under $20 example fifty-five cents, one dollar fifty cents,... Paper and pencil Say: "Write this amount."

A collection of coins for the Give the child a selection of coins. child to count. Say: "Count these and write down the amount. (Continued over)

251 MONEY

TEST MATERIALS TEST PROCEDURES and COMMENTS. ANM.MilmMIN110

Repeat the activities, reading other price tags, writing other amounts from dicta, tion, and recording the totals of other selections of coins, to give the required number of test items within the selectedrange.

Acceptable Response:

The child reads the amounts on the price tags correctly. He writes the amounts, using numerals and the symbols c and $.

Note: If the child attempts to write the amounts in words,ensure that he under stands that numerals and symbols are required.

PROCEDURE:

Ask: "Why do people need money?"or "Why is money used?"or "What would happen if there was no money?"

Acceptable Response:

The child indicates in his answer that he knows that money is used for buying and selling, and that without money a cumbersome barter system would have to be used.

252 1 tMrtliATURE

OBJECTIVES, TEST MATERIALS TEST PROCEDURES and COMMENTS.

The Child,

1, Recognises The child's recognition of differences and changes intemperature can be assessed in the differences in classroom as he discusses and answers questions about temperature,

the weather the seasons the time of day his clothing his physical reaction to heat and cold visits to places with different climates science experiments reading thermometers.

2, Uses vocabulary From observationsas the child takes part in activities as in 1 above, the teacher will note related to familiarity with vocabulary suchas hot, cold, warm, warmer, thermometer, degrees, temperature.

3. Applies knowledge The child's skill in applying knowledge oftemperature in the environment can be in environmental observed in the classroom as he engages it ltivities suchas situations.

making weather records and charts

making graphs involving temperature (See Number, Objective 40.) APPENDIX

254 OUTLINE OF TRIALLING.

FEEDBACK departments, with 136 teachers, which involvedatleast 4,000 childrento This series of goals or objectives, together choose from, and 544 actually tested. with the listsof activities for assessing a child's grasp of knowledge, skills or under- Not all the children tested of course standingsineach, were devised andset were experiencing failure; a number of down in the first instance by the members average and aboveaverage achievers oftheNorth Sydney RegionInfants' were included by teachers so that they Mistresses' Council Committee, and a draft couldfamiliarise themselves with the document wasproduced based onthis methods of testing. work. The text of this document has subsequently been extended, modified or other- At the N.S.R.I.M.C. meeting in 3rd wise amendedinresponseto feed-back term 1977 mistresses present were asked organised by the Committee in order to toindicatetheir willingnessto take involve teachers, and others interested in partinthetrialling scheme and to teaching procedures, in assessment of the commit their schools to participate in usefulness and practicability of the tests. the project. They were asked also to ascertain if the Primary Department of Feedback wasorganisedto come from their school would be willing to co- seven sources. These were: operate, with teachers of years 3 and 4, and if so to inform the committee. 1. Trial lingofthetestproceduresin schools. Followingthis,threecategoriesof 1.Discussions withtrialling teachers in schools were selected, to follow some- in- service groups. what different programmes and 3. Comments from users in the form of schedules: annotationswritteninthetrialling documents concerning the text of the I. Schools with small infants' depart- document. ments; all infants' staff to partici- 4. Responses to Questionnaires returned pate. from all those taking part in the initial II.Larger infants' departments where testing. oneteacherfromeachgrade, 5. Perusal of children's Response Sheets K - 2, would participate. sent to the Committee after testing.' III. Infants' and Primary Departments 6. Assignments written by C.A.E. mathe- where one teacher from each grade, matics students in their third year of K - 4, would participate. training. 7. Comments by and discussions with At a subsequent meeting ofmathematics mathematicians, academics, college lec- advisers, lecturers and co-ordinators, a turers,and others interestedinthis further category was added following aspect of education. an offer by 1)r. R. Perry of Mitchell College of Advanced Education. This Below is an account of the planning and was: resultsfor each of the seven sources of communication. IV. Small schools in the Bathurst area.

1. Triallingof thetestprocedures in Trialling teachers of schools in cate- schools. gories I and II were to be given some (Findings from thistrialling came to in-school familiarization of the docu- the Committee by means of 2, 3, 4 and ment before beginning testing. In cate- 5 in thelist. See below for detailed gory I this was whole-staff discussion discussion of each aspect.) and whole-staff involvement. In cate- goryIIitwas individual discussion The tests were trialled in 42 schools or between teacher and mistress. Trialling 2#2 teachers in categories III and IV were to The particular and detailed findings of be given only a brief letter of explana- their trialling were indicated in com- tion and were to familiarize themselves ments whichwerewrittenonthe with the procedures from the book document pages,_ inanswers to the alone. questionnairesandin responseto personaldiscussionsatthe inservice Principals and Mistresses of the schools courses, all of which are discussed in in the first two categories were to be detail below. invited to an Inservice course at which explanations were given so that they could conductthe familiarization 2. Inservice Discussions. procedures in their own schools and departments. When trialling had been in progress for some time, the special inservice courses Inservice discussions were planned for which had been planned were held; at ather teachers taking part so that they thesethetriallingteachers met and could discuss their progress with the discussedtheir work with committee committee members during the course members. of their trialling. The schedule of these meetings was as When all work from the schools was follows: collated the information was evaluated. Ft was found that there was very little 1st March, 1978.Principals and mistresses difference in results from categories I, of cityschoolstaking part inthe IIand HI. However, of those small trialling were invitedto Naremburn schools which took part, the teachers Inservice Centre. At this meeting it was qad somewhat different findings fry explained that use of the trialling book theother categories. Most said they was to be evaluated under varied condi- found the tests to be both unnecessary tionswhereteachershaddifferent to themselves; and uninteresting to the degrees of guidance. In some schools children. Since there were so few of the teacher would be given the book them (7) and so many of the others with a request that he/she use it after :129) it would seem that their reaction individual reading of the text; in some nay be due to isolation, unfamiliarity schools teacher and supervisor would with the activities used in larger schools discussthetext and plan the pro- and classes, and to pressures of work- cedures; in others there would be whole- oad and time. It would not be possible staff discussion and planning. It was to state whether their responses would explained that reaction from teachers )e characteristic of other small school working under these different condi- situations. tions was necessary because it was felt that the book in its final form would be 'n general, the over-all reactions from purchased and used by hose taking part seemed to be: individual teachers who felt a :a) Teachers recognised the need for need to assist children in their such a series, own classes; :b) they found the sequence useful, :c)they indicated that the instructions principalsor mistresses who generally were clear, and the tests wishedtoinitiatea school easy to follow and to administer, programmeof helpforall d) they found that planning and time children with difficulties. factors caused the greatest problems, e) they located and pointed out areas The book therefore must be useful to wherethey found difficultyor and understood by teachers working ambiguityingiving, marking or alone, and also those working together interpreting tests. in a whole-school context. 243 256 The necessary number of trialling books discussion points. was given out and signed for, so that an accurate recording of returns could be Resultingfromthesediscussions,a kept. number of changes were made. These were mainly as outlined below. The role of the principal or mistress in initiatingandmonitoringthepro- ExtraNotesof explanation gramme in the school according to the were provided. category in which it was placed was explained, and the follow-up (inservice Some instructions were set out for teachers, questionnaires, return of in greater detail. document, etc.) outlined. Additionalprocedureswere 17th April, 1978.Teachers of schools in included where it was felt that Category I and Category II who were activities had been overlooked to carry out the tests were invited to or understandings not catered attend.Discussionof the document for. was held and the role of the teacher explained.Verylittletestinghad Definitions were provided actuallytakenplaceatthisstage. where it was felt these added Kindergarten teachers did not attend to the clarity of the explana- this meeting as it was feltit was too tions (e.g. "oblong"). early in the year for them either to leave new entrants or to begin assessments Ambiguities were removed by of these children's difficulties. using more definite sentence structures. 6th July, 1978.Teachers in Category III wereinvitedtoattenddiscussions. Additional examples were pro- These teachers had had no staff dis- vided where it was felt these cussions at their school. They came to assistedteachers.Sometimes talkabouttheirinvolvement,their further diagrams were included. introductiontothe document, their reactions,theirdifficulties and their The text was broken or retest experiences to date. arranged for easier reading.

19th July,1978. Kindergarten teachers Explanations of the rationale from the Category I and Category II were expanded. schools were invited. Their pupils were now integrated in the schools and they were able to assess the usefulness of 3.Annotations in the Trial ling Document. the early tests in relation to these five- year old children. Teachers trialling the procedures were each given one of the documents, with At all these meetings teachers discussed instructions that it must be returned their findings, reactions, difficulties and to the committee on a specified date misunderstandings. They told of their and that any comments on the text or methods, materials, failures and success- tests would be welcomed. es. They found mistakes and misprints. They criticised weaknesses, indicated Comments wereaskedforonthe strengths, suggested modifications, following: recounted individual experiences and outlined school plans or policies with The, format of the document; regard to testing, assessing or helping thewording of theinstructionsor pupils. questions; the descriptions of the test procedures; Notes were made of all the the availability of certain test materials;

244 5'7 the ease (or otherwise) of construction to, but more detailed than, those noted of other test materials; from theinservicecourses, and the the comprehensiveness of the tests; resulting changes were of the same the reactions or replies of children; nature,i.e.extraNotes,expanded theadaptations of wording used by instructions, additional procedures, and the tester; specific definitions. the non-provision, or further provision, of test items; any difficulties or ambiguities encount- 4.Responses to Questionnaires. ered; any mistakes, omissions or misprints; Questionnaires were sentofallthe anyother matters which thetester teachersin eachcategory;inthe would like to mention. response forms schools were named, but theindividualteacher was not Teachers were askedtowritetheir identified. The questions were set but suggestionsandcommentsonthe under three headings: Interpretation of "facing" pages of the book wherever Text, AdministrationofTests, and appropriate, and to list on the cover General.TheseQuestionnaireswere the page numbers of the text where given out, and returned, at different comments were made. times accordingtothe Category in which the teacher's school was placed. The dates for the return of the trialling Relevant comments from these were documentsvaried accordingtothe extracted and findings collated. category in which the user's school was placed. Questions on the interpretation of the text were answered in detail, also those There was 100% return of books. Each on administration. Those on the general was "marked off' a master list as it was sectionwerenotso productiveof received back by the committee. comment.

The committee studied and evaluated An analysis of the individual questions the comments written for each batch indicatedfairlygeneral agreement in of books as they were received. Some most sections. An indication of the of the books had very few textual types of responses is given. comments; thesegenerally contained casehistoriesorquotedchildren's The firstset of questions concerned answers (either amusing or inconclusive). Interpretation of the Text. Others found misprints and inconsis- tencies. Many suggested better wording What was your reaction to the whole or gave examples of their own changes document on first reading? Why? to the suggested words. Some books Nearly all the respondents express- contained masses of comment, some ed horror, amazement or shock at very few. Often a pertinent comment the thickness of the volume. (In camefromabook with very few its trialling form, its measurements annotations. Great interest in and care- were 31cm x 25cm x 4.5cm.) fulperusualofthe document was evidenced by the detailed and wide- Did your reaction change later? ranging nature of the annotations. The If so, why? length of service of the teacher did Nearly all changed their reactions not seem to have any particular bearing when they actually examined the on the nature of the comments, except contents and discovered the pur- that teachers with the least number of pose of the tests and their compre- years of service generally made more hensiveness, and realised that they sweeping statements. did not have to readitall nor administer all the tests, nor involve The comments were in general similar all the children. 245 258 How clear did you find the explana- easy to follow? tionsinthe introductorystate- Nobody suggested that the format ments concerning of the sheet was hard to follow. (a) Rationale? Did you find it difficult to record Nobody expressed any diffi- theObjectivesbrieflyonthe culty with this section. Child's Response Sheet by using "Key-Words"? (b) Recording? Only one person found it difficult Nearly all found the explana- to select Key-words. This person tions quite clearone or two wanted a suitable list published. did not see the necessity for extensions of the code beyond Did you find the marking code and X adequate? Most triallers found the code suit- Were you able to understand able; a few still preferred just and X. (a) The purpose of the tests from the Objectives stated? Have you any other comments All seemed able to understand concerning the interpretation of the the purpose although a few did whole document, or parts of it? not agree with the necessity for No significant comments emerged, carrying out the tests (as they in fact very few teachers answered knew what thetests would this question. reveal without having to give them). The second set of questions concerned Administration of the Tests. (b) The language used in describing the test procedures? Beforebeginningtesting,what There were no adverse com- familiarizationprocedureswere ments, indeed no very specific adopted? (Independent reading of comments. document? Individualdiscussion with mistress /principal? Whole-staff (c) The language used in setting involvement? Other?) out the Acceptable Responses Most teachersoutlined the pro- from the child? cedures used in their schools; these All seemed to find this clear followed those suggested by the and easily comprehended. A categories of their placement. How- few suggested a bolder kind of ever a surprising variety of instruc- print for clarity or for catching tionswasgiventoteachersin the eye. Category I, in spite of the fact that theprir.v.palsand mistressesall Was there much need to adapt the received the same briefing. wording in the test items to suit individual children, or to clarify What classroom organization was their responses to the test questions? required for administering the tests? Many teachers admitted to chang- Teachers outlined their strategies ing the wording either to agree with forfindingtimetotest. These their own mode of presentation or included team-teaching, withdrawal, because the suggested mathematical relief by mistress, and utilization languagehadnotbeentaught. of library, assembly, play, scripture Many of the changes recorded were or resource periods. They described not significant. the advantages and disadvantages of each. Some used group-work situa- Did you find the format of the tions,butcommentedonthe suggested Child's Response Sheet adverse effect of distraction.

246 259 How were childrenselected for Was there an Objective in which a diagnostictesting? (Low marks? number of the children you tested Constant failure? Lack of progress? continually jailed in the test items? New pupil? Other?) Can you suggest a reason for this All the above were used. commonfailure?(Nottaught? Difficult concept? Other?) Were average and bright children Most teachers did not find such an tested in order to gain familiarity objective. Those who did, indicated with the test procedures? Was this (a) they had not taught the subject beneficial? matter or (b) the test was beyond Many selected bright and average the child's grade level. children so that they themselves could learn the test procedures. Were the test materials suggested readily available or easily made, How long (ingeneral) did you and suitable for the purpose? spendwitheachchildbefore Mostteachersagreedthatthe arriving at a diagnosis? materials wereingeneraleasily Comments varied concerning the found in schoolrooms. Difficulties length of time taken. Sometimes, were mainly with the beam balance. particularlywhereachildwas (A suitable,accuratebalanceis testedinthe afternoons, fatigue difficulttoobtain commercially causeda postponement. A few evenfromeducationalstores.) took as little as ten minutes, some Teachers' aides were often employ- about an hour. The general average ed advantageously in assembling, seemed to be 20 - 25 minutes. Tests organising or manufacturing equip- in measurement took longer. When ment. students tested, they found many procedures could not be given, as Have you any other other com- the work had not previously been ments concerning the administra- treated in the classroom. tion of the tests? Very few teachers responded to How many Objectives (in general) this; those who did re-iterated the had to be tested before arriving at difficulty of arranging time slots or a diagnosis? finding quiet places for testing. The usual response was four, or five, once a starting point was decided up- (Mistress/Principal to answer. ) on. Occasionally teachers started at When you re-tested a child prev- the beginning as a matter of course, iously tested, in order to ascertain but usually "early" starting points reliability of test procedures, were were for the purpose of winning a your results similar? child's confidence and ensuring that Most agreed that the results were he was comfortable in the situation. similar if the re-test was given a short time after the previous test How did you select the objective at before further learning took place. which to commence testing a child? In a few instances the results were At the level of the rest of the class? betterperhapsbecauseof the At the level of a suspected weak- previous test experience, or increas- ness? At the beginning or at a level ed confidence, or the greater skill too easy, to ensure confidence? of the tester. Answers spreadoverthethree methods fairly evenly. (Mistress /Principal to answer. ) How would theresults of this How successful was your method test be useful to you in school of selection? organization? (Classplacements? Most teachers seemed satisfied with Disciplinarypolicies?Workof their selection. resourceteachers?Purchaseof

247 equipment? Other?) edor suggestedhere.Inother Most replied that the chief benefits papers received, a few suggestions wereincurriculum planning, in were offered for shorter screening programme making, in understand- tests. Mg children's needs and in high- lightingequipmentneededfor After arriving at a diagnosis from activities. study of a child's Response Sheet, were you able to arrive at a con- The finalset of questions was of a clusion about the level of General nature. achievement, or his difficulties? All those who replied (about 3/4 of Do you consider that test series those received) had no difficulty, such as this are necessary? Useful? andthis was borne out by the Most respondents considered the detailed diagnosis statements on the tests useful in that they enabled Response Shee ts. teachersto understand and help pupils. The list of ordered objec- Were you able to make a decision tives helped them evaluate their ineach case about the kind of teaching sequences. follow-up work required? All thosewhodiagnosedthe Did you find the knowledge gained difficulties said they were able to from the testing valuable to you in suggest a suitable follow-up pro- terms of gramme. This also was borne out bythedetailedfollow-up plans (a) yourunderstanding of the written on the Response Sheets. child' Many did not respond; of s.itose who did a number The responses from the Questionnaire said they already understood led to some changes in the text of the thei': pupils well enough not to document. Further ambiguities in the need any tests for this purpose. tests were discovered and corrected. A Others said they gained insights few omissionsin sequence came to into the child's thinking not light and were corrected. Wherever a possible before (and not only teacher's misunderstanding was shown mathematical insights). to have arisen from a too brief statement in the text this statement was (b) advantages or benefits for the expanded. Occasionally it was felt that child? There were not many further explanation was needed in the repliestothis;those who Introduction;additional paragraphs answered said that the child were inserted to cover points raised. gained confidence and importance from the testing situation, and established a better 5.Children's Response Sheets relationship with his teacher. At the end of the trialling period for This series of testsis based on each category, the children's Response evaluation of stated objectives. Do Sheets were returned with the annotat- you feel "comfortable" with this ed Trialling Document. These Response approach? Sheets were scrutinized both as to the All except one of those who replied useofflipmarking code and the said they felt comfortable with this teacher's comments on the tests. The approach. No reason was given for diagnosis recorded, and the follow-up the exception. suggested, where included, were also studied for information on teachers' If not, what approach would you attitudes and findings. consider more appropriate? No other approaches were mention- The Response Sheets received showed by the ticks and crosses that the testers Number section. Activities they carried had carried out the procedures faith- out were narrated in detailed form, and fully, and the comments on the child- their findings set out at length. ren's performances further revealed the teachers'carefulobservationof the The assignments of the College students, children as they made their responses. which included their written statemn The Committee realised from scrutiny their questionnaires and response of these Response Sheets from anony- were sent on to the Committer. mous children how vital the comments gave new insights into the trialling column is for expanding or interpreting a different point of view becaus, the verdict of the ticks and crosses. thestudents'youthfulenthusiasm, literalapproachtoinstructions and The type of class in which the child was complete honesty in judgement. They placed (e.g. parallel, graded, ....) did were not dulled by years of routine not appear to have any bearing on the expectations. They took their mathe- general results, nor did the length of matics andtheir theory of teaching the teacher's experience. In all cases the seriously, and they had no "axe to child was tested by his own teacher, grind" in stating their verdicts, since except wherethemistress re-tested, the children they tested were not the or a student carried out an assignment. childrentheyhadtaught (or were supposedto have taught).They No teacher, apparently, had difficulty in practised in a range of school types finding keywords. Although these varied from inner city to affluent outer sub- considerably all were appropriate, and urbs and they included migrant children easily led to the recall of the complete in their samples. statement of the objective. Aspects emerged which were different Excellentdiagnosticstatements were from thosenotedbyestablished made, and where follow-up was indi- teaches in schools. cated this was appropriate and thoroughly planned. No useful modifi- The students' comments revealed cer- cationswere suggested. One person tain attitudes amongst classteachers suggestedthatcompleteobjectives, (e.g.adefinitebiasagainstcertain togetherwithkeywords,should be strands), and also showed the Com- printed in a blank book format. For mittee variations and differing inter- obvious reasons this is impractical and pretations possible in the instructions wasteful a whole book is not needed for certain procedures. to record tests for about half a dozen or so objectives at a particular level. The Notesof explanation on evaluation suggestion that pads of blank response- policy and purpose were added because sheets could be made is much more of someof their comments. Where sensible and useful. instructions were found to be inexact these were rewritten.

College Students' Assignments. 7.Comments from Academics. Two Colleges of Advanced Education (Alexander Mackie C.A.E., and Sydney Two meetings were held with academics Teachers' College) were involved and involvedinmathematicseducation, seventy-five College students took part one beforethetriallingperiod and in the trialling all had had sufficient one after its conclusion. Some of these training to be aware of the concepts peoplehadtakenpartintrialling and activities involved and to be compre'ddiesq of their own in the interven- petent in organising them. Students of ing 'period. At both these meetings S.T.C. trialled the Measurement section relevanrconunents werenotedfor and of Alexander Mackie C.A.E. the later assessment and possible inclusion 249 262 intherevised document. Comments N.S.W. are taught, and expected to use, from the second of these meetings were line graphs only (not "pie" or other area particularlynumerous.Inaddition, graphs); they are shown the base line to "review" copies of the document were allow for easy interpretation, and to sent to other academics too far away to lead on to t: .e notion of the two axes. attend; some of these people returned While it would be possible to construct written comments, which were studied a legitimate line graph without a base and noted by the committee. line,it would be very difficult for a child to interpret or find other informa- Letters from academics, mathematics tionfromit.But in any casethe lecturers,inspectors,mistressesand purpose of the test is to find where a class teachers were also received and child is failing with what he has been considered. The comments fromall taught (if he has been tough) and is not these sources were most valuable and to discover the creative or 'rinovative ledtomanyalterations,ensuring child. greater clarity, consistency and mathematical accuracy in the text. * The word "train" to describe a line of rods all of the same colour.The rod vocabulary used in N.S.W. Department Discussion Points .definitions. of Education publications is used in this book. Adaptations may be made if Some matters raised but not acted upon other terms are used. were from peopleliving outside N.S.W. (sometimes outside Australia) and obviously working from a different syllabus. Since the Discussion Points rationale tests are not designed to find the mathemat- ically mature, but are to assist those who are Other points were brought up, involving failing in what they have been taught, the policies and underlying principles. syllabusfromwhichthechildrenhave worked has a bearing on the tests given, and All are of a general nature and are the result whereadifferentsyllabusisinvolved, of the careful, interested, detailed appraisal modifications would have to be made. The of the reviewers of the document. The following matters are of this type. committee members were grateful :or the opportunity to consider and discuss these * Zero and nought, rectangle and oblong. matters.After duedeliberationitwas These terms relateto convention or decided to make no alteration to the text definition. The definitions followed here either in explanations as already given or are those stated in the present N.S.W. procedures as already described, other than syllabus of mathematics. toindicate here the matters which were discussed and the committee's attitude to * Mass and Weight.In N.S.W. there is an them. attemptto show thedistinction between these, and to use the word * That there is too much emphasis on "mass" correctly in the early stages, so sets. that later the word "weight" may be Itispointed outthat thereis no understood. emphasis on sets for the sake of sets, as in Set Theory, merely an emphasis on * Rod action for subtraction.Test pro- things as they group and re-group in cedures are based on the N.S.W. method theenvironment. The language used of representing this operation (one rod in connection with sets is the most covering all or part of another). Where efficient way vc describing these actions. some other system of rod representation See page 23. is used with Cuisenaire (or other) rods, modifications would have to be made. * Thatthereisthepossibilitythat teachers may give children the rod tests * Graphs: use of a base line.Children in even though they have not worked with 250 263 ods, thus causing trauma or disadvan- "numberstory"or"equation"be age to the children. substituted.) le intended application of the tests sclearlystated. The purposeisto The term "number sentence" is that liscover whether the children under- used and recommended in the N.S.W. tand the work they have been doing mathematics syllabus. "Number story" vith rods and how it is related to the is unsatisfactory as it could be either a :nvironment. If children who have no childish substitute for a correct term, :nowledge of rods are given the tests, or an ambiguous variant to describe the he results will merely confirm this. problemnarrative. "Equation" could not include the use of "greater than", That there are no tests of attitudes. "less than" and "is not equal to" in t was felt that specific tests of attitudes describing number relationships. supposing the committee could devise hem) would not be of great use for * That only certain fractions should be :lassroomteachers. Comments from included, e.g.halves, quarters,thirds, rialling teachers highlighted the fact and that all others should be excluded hat the one-to-one face-to-face situa- or reserved for "advanced" children. tion did in fact reveal a great deal about Ideally, if a child understands the con- attitudesthat was extremely useful cept of fractions as such, the number of :o the teacher in less obvious and more "parts" does not materially affect the iubtle ways, and led to modifications relativedifficultyofthedifferent n the way a child was managed in fractions. By including tests of each ;lass. category of fractions the teacher may ascertain whether the child understands That the complete series of tests is long, the whole concept, or whether only zndthatshorter samplings of the certain common fractions are known. 9bjectives could be given, which would A decision can then be made as to the either give a quick idea of a child's desirabilityof workingwithother grogress, or act as a screening device. fractions,basedonthedegreeof It is again stressed that the tests are not maturity of the child, or on the use lesigned tofind out who isfailing. such knowledge will be to him. They are designed to be used with a child who is known to be failing, to find * That topics are included which are only out where he isfailing, and possibly suitablefor"able" or"advanced" even why he isfailing, To find the children. particular detail in which a child fails It is not intended that children be given will require detailed tests. tests of work beyond that suited for their grade level, nor is it intended that That some of the Objectives are not the tests be used to find, or extend, set down inthe correct order. (A "able" and "advanced" children. The variety of suggestions for correcting or work asset down forms part of a improving the order was given, e.g. 6 learning sequence. It must be remember- after 9,15 before 7.) ed that topics suitable for "advanced" children in one grade, may be (and It was stated in the introductory notes usually are) the topics for the "average" thatthe Objectives are not listedin children in the next grade. Hence it is strictly hierarchical order, because it is not possible to state in the descriptions impossible to arrive at such an order, of the procedures whether they are and in any case all children do not suitable for "advanced" children only, progress through the Objectives in an unless age and grade are also to be identical order. specified for each procedure. Since the tests are designed so that the tester may That the term "number sentence" be follow the progress of an individual replaced. child it would be undesirable to state (Itwassuggestedthattheterms definite grade levels. (The suggestions

251 264 1,11 are given on page 16 are sugges- disappear. This has already been covered tions only but are designed to protect in the introductory notes. young children against being subjected to tests designed for older children.) * That teachers may not be clear as to the focus of importance in Objective 1; * Thattheteststaketoolongto is it Attributes in general, or is it certain administer. attributes only(asspecified inthe Ithas been admitted that one-to-one description) or is it knowledge of the testsaretime-consuming,butthis vocabularyemployedindesignating cannot be avoidedif a child needs the attributes (e.g. the words "blue", individual help; however this is a small "rough", "round" etc.)? price to pay if it means that the success Actually there are three levelsto be of achild'smathematicalfutureis therebyassured.One commentator discerned in this Objective. complained that it took 35 minutes to i. All objects, animate and inanimate, give ninetests!This appears to the have certain properties or character- committee to be very speedy testing istics which we call attributes. The indeed!Timetakenindiagnostic Objectiveis designed inthe first testing now should save much time in instance to ascertain if the child is future remedial planning, teaching and aware of this (that all objects have testing. attributes), and also of the fact thata whole range of attributes * That the language of the tests should be exists which can be applied (when standardised to help teachers. (This was appropriate) to the objects he sees so that teachers when making adapta- around him. (A particular attribute tions to the text do not "cue children can be applied toa number of by suggesting an answer.) differentobjects,e.g.wooden spoon, wooden box, wooden block.) Adequate notes regarding prompting are included. (See page 21.) It is felt that ii. Since obviously the whole range of standardization would be most undesir- possible attributes cannot be taught able when the object isto probe the and tested,itis necessary that a degree of understanding in individual children. An experienced teacher free to selection be made; it is reasonable adapt the wording of the various proce- that the attributes chosen are those dures may be able to identify a number most useful for the child's present of intermediate levels of understanding and future studies (e.g. those of or achievement. colour, shape, size).

Standardizationof languageisonly iii.Such properties or characteristics necessary when comparison of results must be recognised by the child. isto be carried out. This is not the They are described in words (which purpose of these tests. are usually adjectives, or adjectival phrases). For our "normal" child- * That because there are so many tests, ren (i.e. not deaf, blind or mute) set out in so much detail, teachers will recognition of attributes involves go into a state of "quantity shock" alsoknowledgeof therelevant when presented with the total array vocabulary with which we discuss of testing procedures. suchattributes.Hencethetest Ifteachersfeelthatthey haveto must be conducted usingthese administer all the tests, or that a child words, and the child's understand- has to be subjected to all the tests, such ing must be judged from his know- a reaction would be justified. When ledge of them. In this sense the teachersrealisethatthe quantityis test is a test of the understanding of there to allow for a better selection to certain attribute- words. be made of suitable tests for a particular child, then the state of shock should * That a "standard card index system" would be preferable to the whole-book during the period that the document system of presenting the tests.(The hasbeenprogressivelyrevisedand advantages suggested were (i) the handi- retrialled.The committeefeelsthat cap of book size would be overcome, although in-school and in-service guid- (ii) each test would be on one card ance are most helpful, any intelligent, (perhaps colour coded), obviating the caring, resourceful teacher can use the difficulty in administration caused by document unaided. It is intended that turning over pages, and(iii) kits of the first edition of the book be used materials for the various tests could be primarily by teachersinthe North assembledand packaged withinthe Sydney Region of N.S.W., and that this school,andtherelevantcardsfor furthertrial may leadto additional testing could be stored with the particu- revision, if necessary. lar kits of materials.) * That with children for whom English Teachersindiscussionexpresseda is a second language the book may cause preference for keeping the tests in one difficulties. volume, to prevent loss and facilitate This was discussed at length. It was review. The use of separate cards would recognisedthat there aredifficulties increase the likelihood of loss, which withthesechildren inschools. The causes a lack of continuity, and would following points emerged. destroythepossibilityof frequent over-all review. Availability of the tests The tests may indicate whether t in "card" form as a sole presentation child's difficulty is one of language would thus be undesirable. But if used only, or of mathematics only, or as a supplementary aid the idea has of both language and mathematics. much merit. When materials for a test, or series of tests, have been assembled Theformatand suggested pro- it is time-saving and economical if these cedures may give guidance as to can be kept together for future use. the type of English words which Perhaps indexed stackable containers could be included in a language could be used. Teachers can add to the programme. value of such kits by filing with each kit sets of cards containing the relevant Where a child has too little English tests for that kit (one test per card). ft. valid testing, translation into his own language would be necessary That the book should not be dissemi- before any assessment could take nated without prior inservice training. place; thetests however provide (This was so that there should be no the subject matter to be translated. abuse in using or interpreting the tests.) It is desirable that a teacher give the The purpose of trialling the tests in a tests;a parent unless previously variety of types of schools, and using a teacher could not interpret the teachers who have been given a variety responses. A bi-lingual parent could of degrees of guidance was to ascertain however work in conjunction with whether teachers could undestand the the teacher, translating the ques- purpose and language of the tests from tions and the responses, to allow the text alone and could administer and the teacher toassess the child's evaluate the tests in the way intended, level of mathematical understand- without other assistance. ing.

All the oral and written responses from * That the Geometry sectionisvery thetrialling teachers would seem to limited, nothing is included on direction, indicate that they could. Where there position, orientation, scale, etc. were difficulties in the initial writing Thecommittee membersfoundit these have been, as faras possible, strange that while a number of critics eliminated following teachers' comments considered that some topics were only 253 266 suitablefor "advanced" children and "understands"/"does not understand", so should not have been included (e.g. or even "successful"/"is not successful" fractionsother than halves, quarters thenof course these may be used andthirds; "difficult" vocabulary ; instead. certaintypesof numbersentence such as = 9 6) others suggested * That the length of the descriptions of new and different topics for inclusion. test procedures and acceptable responses Someofthetopicsnominated,it should be greatly shortened. isagreed, would providefascinating This suggestion was made by several activitiesfor children. However since reviewers and was considered carefully the tests are designed to help children by the committee while the document who fail in given work on prescribed was being progressively amended. In tonics. the committee felt they could each case the comment came from an not include innovative subjects which educationist not now working in the very few teachers would have presented. classroom. Although practising teachers The most that could be done here was found the document lengthy, not one to indicate and test ways in which teacheraskedthatthedescriptions prescribedtopics could be developed generally be shortened, nor did any and extended to give children greater teacherpointout how aparticular. understanding oftheirwork,for procedure could be stated more briefly. exampleinthepropertiesof plane figures,orintheinterpretationof In nearly every case where the committee graphs. The non-inclusion of tests for members found an ambiguity, a mis- any particular mathematical topic does interpretationoranuncertaintyin not of course indicate that an innovative interpretation an analysis of the situa- teacher should not develop a course tion showed that it resulted from the for children in that topic. brevity of a description of procedure or materialorresponseandthatthe * That regular recording of "success" or difficulty could be avoided by giving "failure" is not good psychology. (It is more detail in each sentence or adding not clear whether this refers to the further sentences. At discussion sessions effect on teachers or children!) teachers asked questions about words and phrases causing difficulty of inter- Whether or not it is good psychology, pretation,and askedthatthese be it is necessary for a teacher to know expanded so as toavoidthese when a child is failing in certain aspects difficulties.Teachersdidnotlike of his work, and to find out where and instructions such as "Continue in the why he fails. To do this certain tests are same manner using similar material", or given, in some of which he will succeed "Proceed as described above for the and in some of which he will fail. When previousactivity",or"Asksimilar the teacher knows those in which he questions for the other objects, some- failsshe can attempt to diagnose the times substituting longer for shorter" reasons for such failure and plan learn- They asked that in statements like the ing or remedial programmes to ensure first, the manner and the material be future success. Itis not intended that setout; thatinstatements like the the child should be aware of the number second, the procedures be written in of his successes and failures or even that fullagain so that there would be no his responses are being labelled in .this turning back of pages; and that in way. (All that he should realise is that statements like the third, the questions his teacher is endeavouring to help him.) withtheirsubstitutions beactually That thisprocedureofrecording listed. Where considered advisable, this successes and failures should beconie was done. regular for a particular child is certainly not envisaged. If the teacher is helped The descriptions of material sometimes by the substitution of phrases such as caused trouble if briefly stated. When "competent"/"not competent" or materials were listed as "two equivalent

25426 ;ets objects"teachers askedthat answers and partial answers, :he range (e.g. 1 - 10 or 1- 20) be given, it,oldert Avoid et, Is in reasoning Ind asked whether the sets were to be or interpretation. But it is felt that this arranged so as to show their equiva- requirement does not apply here. ence, or were to be in random array. The committee members believe that ifa child was to be asked "Is this the normal, natural, everyday manner of longer or shorter? How do you know?" answering aquestion suchas "How teachers asked for a fuller statement many birds flew away?" would be with 3f the Acceptable Response than "the the number only e.g. "four". This, for :hild responds correctly", and requested young children, would not show lack of anexplanationastowhetherthe understanding, particularly if the answer response if correct was to be given one were given without hesitation. The test- tick or two. ermay,ifdesired,follow up the response by asking "four what?"t In all such cases the relevant sentences elicit the response "four birds". But the had to be expanded, or further sen- child should not be considered wrong tences included. if he does not automatically append the noun. After all, adults in normal conver- Therewereotheroccasionswhere sation do not respond inthis way, teachers expressed no concern and had with nouns after every attribute. For no queries, but the committee members example: "How many eggs will 1 buy?" becameawarewhenobservingthe "A dozen". (Not "a dozen eggs".) annotations on the pages of the text, or the tester's comments on a child's Consider the following: response sheet, that procedure for, "What nationality was Murillo? orthepurpose behind, a particular "Spanish" (Not "Spanish nationality ".) activity had been completely misunder- "What colour cardigans do they wear at stood. Analysis showed that the brevity yourson'sschool?" "Green". (Not of the wording was at fault, and the "Green cardigans ".) steps of the activity were rewritten in "Whatchurchdoeshelivenear?" more detailsothatthey could be "Presbyterian". (Not"Presbyterian better understood. church ".) Thus, although the Committee would If the noun is not automatically attach- prefer that the text be briefer that it is, ed to other attributes, why should we itis not possible to arrange this with- expect it to be attached to numerical out causing difficulties, ambiguities or attributes?Further, during theearly misinterpretations. Teachers demanded stages confusion can be caused in some that statements be full, exact, circumstances by this requirement, for unambiguousandapplicabletoall instance when explaining the reasons eventualities. Ensuring this unfortunate- for,or testing the understanding of, lyinvolvedexpansionofthetext our notation system. rather than pruning into brevity. Considerthisdiscussionaboutthe That single-word numerical answers to numeral 5346. "How many hundreds?" narrative-type problems should not be "How many thousands?" The child may accepted. (It was said that these answers answer "three", or "three hundreds"(3 are "hardly acceptable here and certain- or 300) tothefirst question; these ly would not be in primary or secondary obviouslydonotsignifythe same where lack of units means poor under- amount. He may answer "five" or "five standing".) thousands" (5 or 5000) to the second question, which again are not the same Itisrecognisedthatolder children amount. The child may be led to think working with more complex problems thereare300 hundredsand 5000 needtoaffixnamestoalltheir thousands; itis obvious that misunder- 255 268 standings may arise unless the classroom given without prompting by a young responses are interp 1 consistentl. child, actually shows an intermediate stage between oneword answers and Theabilitytoconvertanumeris tl sentence answers. The child is not so answer intoa sentence ("four" irao much attaching a noun to his numerical "There are four birds" or "Four birds response, as givinganincomplete flew away") is one that is necessary, but sentence in which one part is "under- does not come naturally to children; it stood", for example, "(There were) four will only be possible for them to do so birds", or "Four birds (flew away)". after much training and practice. Such askillisuseful when giving written On thesegroundsthecommittee answers to problems, but will first need believes that the three types of answers to be practised in oral form. are all natural, logical and permissible acco, g to the stage of development Thetypeof answer which includes of the child, and that none should be number and s)un ("four birds"), when considered unacceptable.

269 256 DOCUMENTS USED IN THE TRIALLING PROGRAMME

In preparation for the programme A.After the complete series of Objectives The duplicated and hound form of the were listed and the relevant test pro- testproceduresIn, :leencalledthe cedures were compiled they were typed -trialling document" throughout these and duplicated on foolscap paper, then notes. In preparint; for its trial use in collated and stapled into book form schools, various printed notes giving (31cm x 24cm x 4.5cm approximately) information toteachers, or requiring ready for use by teachers in trialling. information from teachers, were issued At the third term Conference of the toschools. These printed notes and NorthSydneyInfants' Mistresses' formsaretheonesreferredtoas Council in 1977 at the time when the doe 'nd the following book v,as in the course of preparation, section. a report was given to the Mistresses present,andotterstotake part in It is felt that a perusal of these may be triallingweresought. A form was of value to people interested in how provided for sending back to the Com- these test procedures were trialled in mittee after due consideration. A copy theschools,or perhaps how similar of this report, and the accompanying projects may be prepared for trialling. form is shown.

"ZEPORT S. The Dialling will be carried out over a period of at lout two terms. and reporting will be necessary during this period MATHEMATICS EVALUATION PROCEDURES 6 it is the intentinn that a few children univ from each class will be tested; TRIALLING it is anticipated that apprnximately 541 tests per child may be required.

The Procedures and Test Items in Mathematics Evaluation for Number. Shape if you are prepated to have ynur school involved in this Mailing, and to take and Measurement I length, Man. Volume, Temperature. Area, Time and part youttelf, please fill in the attached form, and teturn it to Mrs SI Muir, Mnney) have now been written 26 Orchard Rnad, Beecroft 2119 on or before 1st November, 1977. The neat stage is that of milling When thew (nun, have been received. a selection of sghouls will be made, so that there will be a balanced sampling. You will be informed before the end For a broad anessmentit the document a wide sped+ nth of school notations of this term if your school has been selected, and of the category in which the is required nulling will be required (e.g. Number.SLape. Length). Thew will include staffs on a whole school basis N.S.R.I Mr Committee for individual teachers in staffed wheels la) kindergarten teachers. Mat hemline Evaluation. (b) Year 1 teachers, tcl Year .7 teachers teachen in one teacher schools teachers "(tower peccary classes

Berm carrying out the trials. some teachers will be given the document for study without any prior discussion tome teachers will take pan in individual dist:moon about the document with mistress/principal N.S.R IM.0 toms will part ictpate in whole-stall dtscursion resnons about the Mathematics Evaluation Comminee document Vie would like to be involved in the 'Dulling of the Evaluation Document Trilling will involve the follnwing conditions in 1978.

Department I Test items are tribe given, in the firm instance. to children whose natural School ...... language is English Later. if present in the school, apptoptiate children in the categories shown below may be tested. If desired Phnne Numbet (a) Those with minimal understanding of English (original language. and not English. spoken at home) Addtess (b) Those with fair competency in Englahlortginal language. ono riot English, spoken at home) (c) Thnse with fair competency in English (both languages spoken at Pomade home) The category would be staled, to wnrds, no the Child's Response Sheet Mistress or Pnnctpal (Children with nn understanding of English would nor be subject to testing.) Number of classes in dept...... 2.Meetings and interviews with those partictpating may be held before and Approximate total enrolment of dept. n 1978 after the mailing. .- R.S.VP. by 1.11.77 Signed .... 3 The completion of questionnaires by clan teachers and by formless/ ptractpal writhe required. CO Mts. N. Muir, Date 26 Orchard Road. 4 All documents, including Response Sheets of children tested. are to be resumed by or on the dun nornmated nEECROFT. 2119.

2!0) 257 B.When the offers of participation were of the strands for testing. A list was received schools were placed in one made and duplicated using the following of thecategoriesandallottedone headings: